Engineering Structures 46 (2013) 294–301

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Low-cycle fatigue testing of extruded aluminium alloy buckling-restrained braces

Chun-Lin Wang a,b, Tsutomu Usami b,⇑, Jyunki Funayama b, Fumiaki Imase b

a International Institute for Urban Systems Engineering, Southeast University, Sipailou 2#, Nanjing 210096, Chinab Department of Civil Engineering, Meijo University, Tempaku-ku, Nagoya 468-8502, Japan

a r t i c l e i n f o

Article history:Received 9 July 2011Revised 5 February 2012Accepted 17 July 2012Available online 13 September 2012

Keywords:Seismic designBuckling-restrained braceAluminium alloyLow-cycle fatigueExtrusion manufactureFatigue curve

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.07.016

⇑ Corresponding author. Tel./fax: +81 52 838 2363.E-mail addresses: usamit@meijo-u.ac.jp (T. Usam

Wang).

a b s t r a c t

Aluminium alloys have recently been employed to manufacture buckling-restrained braces (BRBs) withthe aim of improving BRB durability in corrosive environments. Based on the ease with which aluminiumalloys are extruded, the extruded aluminium alloy BRB is proposed to avoid the welded and relativelycomplex BRB end used in previous BRB research. This experiment, including 10 nearly identical speci-mens with or without stoppers, was performed to address low-cycle fatigue performance. According tothe test results, the extruded BRB possessed a stable and repeated hysteretic performance, and the frac-ture location was random in the yielding portion of the brace. The failure of the extruded BRB wasregarded as a brittle fracture compared to the typical failure of a steel BRB. The comparison betweenspecimens with and without stoppers showed that the stoppers had no clear influence on the cumulativeinelastic deformation, provided that the BRB was horizontally placed and the strain amplitude was lowerthan 2%. The low-cycle fatigue damage evaluation formula for the extruded BRB is recommended as a ref-erence for strain-based damage assessment.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction calibrate numerical models. The research to date has accelerated

A rational approach for the seismic design or retrofit of struc-tures is based on the introduction of energy dissipation devices,which act as ductile fuses during an earthquake to minimise thedamage of both structural and non-structural elements. Abuckling-restrained brace (BRB), which is a type of metallic yield-ing-based device, can yield under both tension and compressionwithout buckling and exhibits a stable elastic–plastic hystereticbehaviour. Previous research and engineering applications showthat BRBs improve the seismic performance of conventional build-ings and bridges when they replace diagonal braces.

Experimental and numerical studies at the component andframe levels have been conducted to promote the application ofdifferent types of BRBs. For example, Chou and Chen [1] addressedexperimental and numerical research on a BRB that employed tworestraining systems to restrain the buckling of a core plate. Sabelliet al. [2] investigated the seismic responses of buckling-restrainedbraced frames (BRBFs) with several important parameters. Kimet al. [3,4] and Wu et al. [5] provided design procedures to quantifythe responses of BRBFs. Usami et al. [6] and Chen et al. [7] numer-ically studied the effect of BRBs on the seismic performance of aretrofitted steel bridge. In particular, Mazzolani [8] tested threesub-structures of a real reinforced concrete building that wasupgraded with BRBs and other energy-dissipation techniques to

ll rights reserved.

i), chunlin@seu.edu.cn (C.-L.

the application of BRBs in civil engineering.Recently, there has been a focus on the use of aluminium to im-

prove the durability of metallic energy-dissipation devices in cor-rosive environments. Aluminium oxide is naturally generated onthe surface of a metal and is more stable than pure aluminium.Thus, it can act as a natural film to protect the metal’s body againstcorrosion, which is very important for the successful application ofBRBs in saltwater environments [9].

Besides its excellent corrosion resistance, aluminium alloy of-fers a wide range of useful properties: it is light and has onlyone-third the density of steel, it is easily fabricated, and its recycla-bility provides both economic and environmental benefits. Forthese reasons, aluminium has been employed to develop shearpanels or links as energy-dissipation dampers in some studies[10,11]. Nevertheless, contact between aluminium and steelshould be avoided. Steel bolts should be filmed by fluorine resinduring the assembly of aluminium alloy BRBs. In this experimentalstudy, normal high-strength steel bolts were used for convenience.An economic analysis confirmed that the major costs of this assem-bly were related to the aluminium alloy, but the total cost, includ-ing life cycle costs, was deemed acceptable.

2. Previous research

Although the low-cycle fatigue performance of aluminiumalloys has been widely verified in the literature [12–16], littleattention has been paid to the use of aluminium and its alloys in

(a)

BMAngle

(b)

Weld toe

Fig. 1. Configurations of the (a) welded and (b) bolt-assembled aluminium alloy BRBs [17].

StopperRib

(d)(c)(b)(a)

Die

Semi-finished

Fig. 2. Extrusion process of the aluminium alloy BM: (a) step I; (b) step II; (c) step III; (d) step IV.

190

2015

B(Unit: mm)

30

9

240 87.5 L=1360 87.5 240

190

180 45 1565 45 180

190

190

12

(a) (b)

t

t

RibStopper

(c)Fig. 3. Extruded BM (a) cruciform section; (b) brace member; (c) photo.

C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301 295

the manufacture of BRBs. As a result, there are few detailed testdata on aluminium alloy BRBs. In the past 2 years, one group ofauthors [17]has conducted several experiments on welded andbolt-assembled aluminium alloy BRBs, whose configurations areshown in Fig. 1.

According to the test results, the low-cycle fatigue life of thewelded BRB was remarkably influenced by the welding of the ribs,which were used to improve the out-of-plane stiffness of the unre-strained region of the brace member (BM). Consequently, the bolt-assembled aluminium alloy BRB was proposed to avoid the ribwelds. Further experiments showed that the performance of thebolt-assembled BRB with spot-welded stoppers was affected bythe spot-weld of the stoppers, which were used to prevent theslip-off movement of the restraining members (RMs). Therefore,a bolt-assembled aluminium alloy BRB without stoppers was

subsequently tested, and the corresponding results showed thatit exhibited a stable hysteretic behaviour throughout the durationof the test. The performance of the bolt-assembled aluminiumalloy BRB without stoppers partially satisfies the demands pro-posed for steel high-performance BRBs (HPBRBs) used in bridgeengineering, as suggested by authors [7,18]. The HPBRB is expectedto survive after three times of strong earthquakes without severedamage, and it does not need to be replaced during the typical lifecycle of a bridge.

Although a bolt-assembled BRB has good hysteretic perfor-mance, the configuration of this BRB’s end is relatively complex,as shown in Fig. 1b. Considering that an aluminium alloy is easilyextruded, the extrusion process is employed to fabricate BMs, asshown in Fig. 2. A semi-finished product can be obtained throughthe use of a die on the finished cruciform section, and this

Table 1Geometric dimensions and structural properties of the BMs.

Series Specimens L (mm) B (mm) t (mm) A (mm2) k P0 (kN) d0 (mm)

EA-WS EA-WS-R1 1360 99.9 10.0 999 471 185.4 3.94EA-WS-1.0 99.9 10.1 1009 465 187.3 3.94EA-WS-1.5 100.1 10.1 1011 466 187.6 3.94EA-WS-2.0 100.0 10.2 1020 462 189.3 3.94EA-WS-2.5 99.8 10.2 1018 461 188.9 3.94EA-WS-D1 100.1 10.2 1021 462 189.5 3.94EA-WS-D2 100.0 10.1 1010 466 187.5 3.94

EA-NS EA-NS-R1 100.1 10.0 1001 471 185.8 3.94EA-NS-1.5 100.1 10.1 1011 466 187.6 3.94EA-NS-2.0 100.2 10.1 1012 466 187.8 3.94

Note: L is the nominal length of the BM’s yield portion; B is the width; t is the thickness; A is the sectional area; k is the slenderness ratio on the weak axis; P0 is the Ar0; and d0

is the Le0. The symbols L, B, and t are illustrated in Figs. 3b and 4.

B

t

48 48200

25

25d0=2

Bolt

Restraining member (RM)

Brace member(BM)

(Unit: mm)

d=1

Unbonding material

Fig. 4. Details of the cross-section.

Table 2Chemical compositions of aluminium alloys.

Type Chemical compositions (%)

Si Fe Cu Mn Mg Cr Zn Ti

HS63S-T5 0.57 0.21 0.01 0.02 0.76 0.01 0.01 0.02A6061S-T6 0.08 0.26 0.01 0.02 2.62 0.18 0.02 –

296 C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301

procedure is used to manufacture the BM. It is important that a BMwith ribs and stoppers has no welds, although the extrusion pro-cess has a small negative effect on fatigue performance, as dis-cussed in the literature [19].

Based on previous research of extruded aluminium alloy BRBs,an experiment involving 10 specimens was conducted at the Ad-vanced Research Centre for Seismic Experiments and Computa-tions (ARCSEC) of Meijo University to address their low-cyclefatigue performance and to evaluate the effect of stoppers. Detailsof the test program and results are summarised below.

3. Test program

3.1. Dimensions of specimens

Fig. 3a and b give the nominal dimensions of the die’s cruciformsection and the finished BM. As shown in Fig. 3c, there is no weld inthe BM, but it has stoppers at its centre and ribs in its unrestrainedregion. Table 1 lists the measured dimensions and structural prop-erties of the BM. As shown in Fig. 4, the BM was inserted between apair of RMs, and the unbonding material, a type of butyl rubberwith a thickness of 1 mm, was used to minimise friction. Gaps withwidths d = 1 mm and d0 = 2 mm were provided between the BMand the RMs.

95

28@150

50

4848

200

25

Fig. 5. Dimension

Aluminium alloy flat plates were selected for manufacturing thespecimens’ RMs by considering the advantages and disadvantages,such as lightness and contact corrosion with other materials, ofvarious aluminium alloys. The nominal dimensions of the RMsare shown in Fig. 5. The RM dimensions employed in the literature[17] were used again in this experiment, and the overall bucklingof the specimen and the relative displacement between the twoRMs were not observed in any of the tests. The frictional force be-tween the two RMs was relied upon to prevent relative displace-ment because each high-strength bolt was tightened by anelectric wrench with a torque of 170 N�m.

3.2. Adopted material characteristics

A new aluminium alloy, HS63S-T5, provided by Nippon LightMetal Company, LTD., was used for the BM. This material was devel-oped to increase the strength of the A6063 aluminium alloy. AnA6061S-T6 aluminium alloy was employed to fabricate the RMs.Table 2 lists the chemical compositions of two materials, and Table3 lists the data of the coupon tests performed on the materials.Fig. 6 presents the stress–strain curves of the HS63S-T5 andA6061S-T6 alloys obtained from coupon tests, which show thatthe A6061S-T6 has a notably higher yield stress than the HS63S-T5.

3.3. Labelling

Table 1 lists the tests specimens. The stoppers of the EA-NS-R1,EA-NS-1.5 and EA-NS-2.0 specimens were ground to evaluate theireffects on the BRB’s performance. Each specimen was labelled sothat the type and the testing protocol could be clearly identified.

500

7575

200

50

(Unit: mm)

s of the RMs.

Table 3Material properties of aluminium alloys.

Alloy Type E (GPa) r0.2 (MPa) r0 (MPa) e0.2 (%) e0 (%) ru (MPa) eu (%) m Objective

HS63S-T5 64.9 206.3 185.6 0.52 0.28 230.3 8.02 0.35 For BMA6061S-T6 72.1 273.8 246.5 0.58 0.34 300.9 7.82 0.33 For RMs

Note: E is the initial Young’s modulus; r0.2 is the 0.2% proof stress; r0 is the 0.9 r0.2; e0.2 is the 0.2% proof strain; e0 is the deformation corresponding to the stress r0; ru is theultimate tensile strength; eu is the tensile strain at fracture; and m is the Poisson ratio.

0 1 2 3 4 5 6 7 80

50

100

150

200

250

300

350

HS63S-T5

Stre

ss σ

(M

Pa)

Strain ε (%)

A6061S-T6

0.0 0.1 0.2 0.3 0.4 0.50

50

100

150

200

250

300

Fig. 6. Stress–strain curves from the coupon tests.

JacksSpecimen

Displacement

transducers

Data acquisition system

Fig. 7. Testing setup.

C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301 297

The first part of the label indicates the type of the BRB, where ‘‘EA’’refers to the extruded aluminium alloy BRB; the middle part indi-cates whether the BRB contains stoppers, where ‘‘WS’’ and ‘‘NS’’ re-fer to specimens with and without stoppers, respectively; and thelast part indicates the strain amplitude of the testing pattern,where ‘‘1.0’’ refers to a constant 1% strain amplitude and ‘‘R1’’,‘‘D1’’, and‘‘D2’’ indicate the variable strain amplitude, which is ex-plained in the following section.

ε0

Δε/2

−Δε/2

δ0

−δ0

0.5ε0

−0.5ε0−ε0

−2δ0

−3δ0

−4δ0

2δ0

3δ0

4δ0

n

0 0

(a) (b)Fig. 8. Testing patterns: (a) stepwise incremental strain amplitude (R1); (b

3.4. Testing setup

As shown in Fig. 7, an extruded specimen was horizontally pin-ned between two rigid pillars. The force was applied by two paral-lel jacks. The axial displacement of the restrained yield portion wasmonitored by eight displacement transducers, which weremounted on both ends of the specimen.

3.5. Testing protocol

The experiment employed several tensile and compressive re-versed cyclic testing patterns controlled by the axial strain of thespecimens. As shown in Fig. 8a, the first testing pattern (R1) wasa stepwise incremental cyclic protocol adopted in the EA-WS-R1and EA-NS-R1 specimen tests. The strain amplitude of each cycleincreased from e0 to 10 e0 (approximately 2.8%) with the incremente0 and then maintained this amplitude until failure.

As shown in Fig. 8b, the second testing pattern included twostages. The first stage was composed of one 0.5 e0 strain amplitudecycle and one e0 strain amplitude cycle, which were used to eval-uate the equipment system. In the second stage, the constant strainamplitude specified in Table 4 was imposed cyclically until thespecimen failed.

The third testing pattern (D1 or D2) was used to directly evalu-ate the damage index based on Miner’s Law. As shown in Fig. 8c,two cycles of the e0 strain amplitude were first imposed as an eval-uation procedure. In the testing of specimen EA-WS-D2, the im-posed strain amplitude was 1% (4 cycles), 2% (4 cycles) and 2.5%(employed until specimen failure), where as in the testing of theEA-WS-D1, the imposed strain amplitude was 1% (8 cycles), 1.5%(2 cycles), 2% (1 cycle) and 2.5% (employed until specimen failure).

Because the engineering strain, e, is defined as the relative dis-placement divided by the original length of the BM’s yield portion,the strain control becomes equivalent to the displacement controlduring the testing. Thus, these tests are conducted by controllingthe axial displacement.

4. Test results

4.1. Stress–strain relationships

Fig. 9 provides stress–strain curves for the extruded specimens.The tensile states of the BRBs are displayed in the positive

δ0

−δ0−1%−2%

−2.5%

1%2%

2.5%

2

n1

n2

0

(c)

) constant strain amplitude; (c) variable strain amplitude (D1 or D2).

Table 4Test results of the specimens.

Series Specimen De/2 (%) De (%) Nf ni CID (%) D Failure position

EA-WS EA-WS-R1 – – – – 31 1.14 MiddleEA-WS-1.0 1.0 2.0 40 – 93 1.40 LeftEA-WS-1.5 1.5 3.0 10 – 42 1.22 MiddleEA-WS-2.0 2.0 4.0 5 – 30 1.41 RightEA-WS-2.5 2.5 5.0 2 – 17 1.01 MiddleEA-WS-D1 1.0 2.0 – 8 49 1.88 Middle

1.5 3.0 – 22.0 4.0 – 12.5 5.0 – 2

EA-WS-D2 1.0 2.0 4 41 1.72 Right2.0 4.0 42.5 5.0 1

EA-NS EA-NS-R1 – – – – 34 1.45 MiddleEA-NS-1.5 1.5 3.0 9 – 37 1.09 MiddleEA-NS-2.0 2.0 4.0 6 – 36 1.70 Middle

Note: De/2 is the strain amplitude; De is the strain range; Nf is the number of failure cycles; ni is the occurrence frequency according to Dei range; CID is the cumulativeinelastic deformation; and D is the damage index.

298 C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301

direction. The specimens demonstrated stable and repeated hys-teretic curves at all of the testing amplitudes, even though themaximum strain amplitude was 2.5%.

The hysteretic behaviour of some extruded specimens testedwith comparatively large strain amplitudes, such as EA-WS-2.0

-300

-200

-100

0

100

200

300

Failurepoint

EA-WS-1.0

σ (M

Pa)

Nf=40 -300

-200

-100

0

100

200

300EA-WS-1.5

Nf=10

σ (M

Pa)

EA-WS-R1 EA-WS-D1

-3 -2 -1 0 1 2 3

EA-NS-R1

ε (%)

EA-NS-1.5

-300

-200

-100

0

100

200

300

σ (M

Pa)

-300

-200

-100

0

100

200

300

σ (M

Pa)

-300

-200

-100

0

100

200

300

σ (M

Pa)

-300

-200

-100

0

100

200

300

σ (M

Pa)

-3 -2 -1ε (

-3 -2 -1 0ε (

-3 -2 -1 0 1 2 3ε (%)

-3 -2 -1 0 1 2 3ε (%)

-3 -2 -1 0 1 2 3ε (%)

Fig. 9. Stress–strain relati

and EA-WS-2.5, is slightly asymmetric in both tension and com-pression. For example, the maximum compressive stress of theEA-WS-2.0 specimen was 4.5% greater than its maximum tensilestress. In comparison, under the same strain amplitude conditions,the maximum compressive stress of a steel BRB with an identical

-300

-200

-100

0

100

200

300EA-WS-2.0

Nf=5

σ (M

Pa)

-300

-200

-100

0

100

200

300EA-WS-2.5

Nf=2

σ (M

Pa)

EA-WS-D2

Nf=9

EA-NS-2.0

Nf=6

-300

-200

-100

0

100

200

300

σ (M

Pa)

-300

-200

-100

0

100

200

300

σ (M

Pa)

0 1 2 3%)

-3 -2 -1 0 1 2 3ε (%)

-3 -2 -1 0 1 2 3ε (%)

1 2 3%)

-3 -2 -1 0 1 2 3ε (%)

-3 -2 -1 0 1 2 3ε (%)

ons of the specimens.

C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301 299

configuration is 5.7% greater than its maximum tensile stress [20].One reason for the similarity in asymmetry between the steel andaluminium alloy BRBs can be explained as follows. The shear fric-tion occurred between the BM and the RMs when the BM sustainedthe compressive force. This friction was correlated with the frictioncoefficient and the compressive force. Additionally, because thesame unbonding material was used, this similarity was observedin BRB experiments that were identical in every way except forthe choice of material. The first loop in the BRB’s hysteretic behav-iour was hardly affected, and the subsequent loops were influencedby the strain hardening effect. These results are identical to the testresults of steel BRBs [20] and A5083P-O aluminium alloy BRBs [17].

4.2. Fatigue performance

The cumulative inelastic deformation (CID) indicates the cumu-lative ductility deformation capacity excluding the elastic portionbefore the failure of specimens, which is also listed in Table 4.The CID values are greater than 30% when the imposed constantstrain amplitude is lower than 2% or the imposed amplitude isvariable.

The specimens with and without stoppers were tested to evalu-ate the effect of the stoppers on the extruded BRB’s performance.During the test of the EA-WS-R1 and EA-NS-R1 specimens, theslip-off movement between the BM and the RMs was not clearly ob-served, and their CID values were nearly equal. Moreover, the num-ber of fatigue cycles Nf dropped from 10 to 9 during a comparisonbetween the EA-WS-1.5 and EA-NS-1.5 specimens under the same1.5% amplitude, while Nf increased from 5 to 6 during a comparison

EA-WEA-WS-1.0

EA-W

EA-WS-D1

EA-WS-1.5

Fixed End

EA-N

E

EA-NS-1.5

Left Middle

EA-WS-R1 E

EA-WS-2.5 EA-WS-D1

EA-NS-R1 EA-NS-1.5

EA-WS-1.0

Fig. 10. Failure modes of th

between the EA-WS-2.0 and EA-NS-2.0 specimens under the same2% amplitude. Therefore, it was concluded that the stoppers hadno remarkable influence on the low-cycle fatigue performance ofthe extruded BRB when the BRB was horizontally placed and theimposed strain amplitude was smaller than 2.0%.

4.3. Failure modes

Fig. 10 presents the failure mode of the extruded BRBs, andFig. 11 presents the failure modes of the bolt-assembled alumin-ium alloy BRBs and the steel BRBs tested with 2% strain amplitudereported in the relevant literature [17,20]. The rupture of the ex-truded BM induced the specimen’s failure, the locations of whichwere random. Based on the comparison between specimens withand without stoppers, it appears that the stoppers did not signifi-cantly affect fracture. The fracture surfaces of the extruded speci-mens were much rougher than those in the blot-assembledaluminium alloy BRBs. One possible reason for this differencecould be that the defects of the extrusion process, which can beconsidered as an existent structural imperfection, had an influenceon the failure mode. Thus, the failure of the extruded BRBs was re-garded as a brittle fracture and no apparent necking plastic defor-mation occurred before fracture.

Fig. 12 shows the final loops of the corresponding stress–strainrelationships. During the testing of the aluminium alloy BRBs, theactuator’s force dropped rapidly to zero at the instant of eachBM’s rupture when the imposed velocity measured 0.15 mm/s.However, the failure of the steel BRBs in previous studies was con-sidered to be a ductile fracture because a declination process and a

S-R1

S-2.5

EA-WS-2.0

EA-WS-D2

Loading End

S-R1

A-NS-2.0Right

A-WS-1.5 EA-WS-2.0

EA-WS-D2

EA-NS-2.0

e extruded specimens.

Crack

Fracture(a) (b)Fig. 11. Failure modes of (a) a bolt-assembled aluminium alloy BRB [17] and (b) asteel BRB [20].

300 C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301

clear inelastic deformation were observed on the BM, as shown inFigs. 12c and 11b.

5. Low-cycle fatigue life

5.1. Fatigue curves

The Manson–Coffin equation is often used to indicate the rela-tionship between the number of failure cycles Nf and a strain rangeand can be given as [21]:

De ¼ Ce � ðNf Þ�ke þ Cp � ðNf Þ�kp ð1Þ

where De is the total strain range; Nf is the number of failure cycles;and Ce, Cp, ke and kp are constants that depend on the material. Be-cause the plastic strain of the BM is much larger than its elasticstrain and the total strain range is directly measured from theexperiment, Eq. (1) can be approximately given by:

-3 -2 -1 0 1 2 3-500

-400-300

-200-100

0

100200

300400

500

σ (M

Pa)

ε (%)

Failure point

(a)

-500

-400-300

-200-100

0

100200

300400

500

σ (M

Pa)

-3 -2 -1ε

Fig. 12. Final loops of the stress–strain relationships of (a) the EA-WS-2.0 specime

1 10 1000.01

0.02

0.03

0.04

0.05

0.060.07

: EA-WS-1.0~2.5

: Equation (3) : Equation (4)

: EA-WS-D1: EA-WS-D2: EA-NS-1.5: EA-NS-2.0

Stra

in r

ange

Number of cycles (Nf)

Δε

Fig. 13. (a) S–N curves for the extruded

De ¼ C � ðNf Þ�k ð2Þ

Based on test results of the extruded BRBs with stoppers under con-stant strain amplitude (from EA-WS-1.0 to EA-WS-2.5, as listed inTable 4), the values of C and k are obtained by the least mean squaremethod. The Manson–Coffin equation for the extruded BRB can beexpressed as:

De ¼ 0:063 � ðNf Þ�0:306 ð3Þ

As shown in Fig. 13a, some specimens fall below the fatigue curve(S–N curve) of Eq. (3), which suggests that the extruded BRB is un-safe for structural applications. Combined with the standard errorobtained by the least mean square method, the recommended Man-son–Coffin equation for the extruded BRBs is updated to obtain:

De ¼ 0:060 � ðNf Þ�0:329 ð4Þ

To present the test results of the specimens under the variablestrain amplitude with S–N curves in Fig. 13a, the equivalent strainrange Deeq and the equivalent number of fatigue cycles N can beexpressed as [22]:

Deeq ¼PðDe1=k

i � niÞN

!k

ð5Þ

N ¼X

ni ð6Þ

Fig. 13a shows that the specimens tested with the variable strainamplitude lie over the recommended S–N curve according to Eq.(4). Fig. 13b compares the fatigue curves of the different BRBs thathave been recently studied [17,20]. The low-cycle fatigue perfor-mance of the extruded aluminium alloy BRB is better than that of

(b)

Failure point

-500

-400-300

-200-100

0

100200

300400

500

(c)

σ (M

Pa)

Failure point

0 1 2 3 (%)

-3 -2 -1 0 1 2 3ε (%)

n, (b) the bolt-assembled aluminium alloy BRB [17] and (c) the steel BRB [20].

1 10 100

0.01

0.02

0.03

0.04

0.050.060.070.08

Welded BRB

Bolt-assembled BRBSteel BRB

Extruded BRB

: Extruded BRB : Welded BRB : Bolt-assembled BRB : Steel BRB

Stra

in r

ange

Δε

Number of cycles (Nf)

Aluminum alloy

BRB; (b) comparison of S–N curves.

C.-L. Wang et al. / Engineering Structures 46 (2013) 294–301 301

the welded aluminium alloy BRB, but it is lower than that of thebolt-assembled aluminium alloy BRB or the steel BRB.

5.2. Miner’s law

The damage caused by one cycle at the Dei strain range is de-fined as Di = 1/Nfi, where Nfi is the number of failure cycles forthe Dei strain range [9]. Thus, the cumulative damage throughoutthe strain–time history can be expressed as:

D ¼X ni

Nfið7Þ

where ni is the occurrence frequency at the Dei strain range, and D isthe cumulative damage index. The relationship expresses the lineardamage rule proposed by Miner for the prediction of aircraft fatiguelife [21]. When the cumulative damage reaches 1.0, the structuresuffers fatigue failure. Based on test results of the specimens witha constant strain amplitude, the D values of the EA-WS-D1 andEA-WS-D2 can be given directly as 1.6 and 1.4, respectively. Bycombining Eqs. (2) and (7), the relationship between the cumulativedamage index and the strain range can be given as:

D ¼ C�1=k �X

ni � ðDeiÞ1=k ¼ C �X

ni � ðDeiÞm ð8Þ

Thus, the cumulative damage formula for the extruded aluminiumalloy BRB is given as follows:

D ¼ 5:1� 103 �X

ni � ðDeiÞ3:04 ð9Þ

The test results were used to quantify the validity of Eq. (9). Thecumulative damage index is given in Table 4 and indicates that thisevaluation formula for extruded BRBs is conservative and effective.

6. Conclusions

This paper further investigated the potential for the use of easilyextruded aluminium alloy BRBs in structural applications. Tests ofextruded aluminium alloy BRBs with and without stoppers wereperformed to understand their low-cycle fatigue performance.The main results are summarised as follows:

(1) Low-cycle fatigue tests show that extruded BRBs exhibit sta-ble and repeated hysteretic performance and that the CIDvalue is larger than 30% when the imposed constant strainamplitude is lower than 2% or when the specimen is undervariable strain amplitude.

(2) The fracture location of extruded BRBs is random in the yieldportion of the BM. Its failure is regarded as a brittle fracturein comparison to the failure of steel BRBs.

(3) A comparison between the specimens with and withoutstoppers shows that the stoppers have no clear influenceon low-cycle fatigue performance when the BRB is horizon-tally placed and the imposed strain amplitude is lower than2%.

(4) The low-cycle fatigue damage evaluation formula forextruded aluminium alloy BRBs is recommended as a refer-ence for strain-based damage assessment.

Acknowledgement

The study is supported by JSPS Grant-in-Aid for Scientific Re-search (B) 23360200 and in part by grants from Japan Scienceand Technology Agency for ‘‘Evaluation and Mitigation of Environ-ment Impacts of Earthquake and Typhoon Disaster on Urban Areaand Infrastructures’’ (Project Title: Refined Analysis and DamageControl of Earthquake Disaster Impact on Bridge Structures), underthe Strategic Japanese-Chinese Cooperative Program on Scienceand Technology (S&T) for Environmental Conservation and Con-struction of a Society with Less Environmental Burden.

References

[1] Chou C-C, Chen S-Y. Subassemblage tests and finite element analyses ofsandwiched buckling-restrained braces. Eng Struct 2010;32:2108–21.

[2] Sabelli R, Mahin S, Chang C. Seismic demands on steel braced frame buildingswith buckling-restrained braces. Eng Struct 2003;25:655–66.

[3] Kim J, Choi H. Behavior and design of structures with buckling-restrainedbraces. Eng Struct 2004;26:693–706.

[4] Kim J, Seo Y. Seismic design of low-rise steel frames with buckling-restrainedbraces. Eng Struct 2004;26:543–51.

[5] Wu J, Liang R, Wang C-L, Zhou Z. A modified capacity spectrum method withdirect calculation of seismic intensity of points on capacity curve. J Earthq Eng2011;15:664–83.

[6] Usami T, Lu Z, Ge H. A seismic upgrading method for steel arch bridges usingbuckling-restrained braces. Earthq Eng Struct Dyn 2005;34:471–96.

[7] Chen X, Ge H, Usami T. Seismic demand of buckling-restrained braces installedin steel arch bridges under repeated earthquakes. J Earthq Tsunami2011;5:119–50.

[8] Mazzolani FM. Innovative metal systems for seismic upgrading of RCstructures. J Constr Steel Res 2008;64:882–95.

[9] Mazzolani FM. Aluminium alloy structures. 2nd ed. London, UK: E & FN Spon;1995.

[10] De Matteis G, Mazzolani FM, Panico S. Experimental tests on pure aluminiumshear panels with welded stiffeners. Eng Struct 2008;30:1734–44.

[11] Sahoo DR, Rai DC. Seismic strengthening of non-ductile reinforced concreteframes using aluminum shear links as energy-dissipation devices. Eng Struct2010;32:3548–57.

[12] Coffin Jr LF. Low cycle fatigue: a review. Appl Mater Res 1962;1:129–41.[13] Brown GW, Ikegami R. The fatigue of aluminum alloys subjected to random

loading. Exp Mech 1970;10:321–7.[14] Chung YS, Abel A. Low cycle fatigue of some aluminum alloys. In: Solomon HD,

Halford GR, Kaisand LR, Leis BN, editors. Low cycle fatigue. ASTM STP 942,Philadelphia: American Society for Testing and Materials; 1988. p. 94–106.

[15] Borrego L, Abreu L, Costa J, Ferreira J. Analysis of low cycle fatigue in AlMgSialuminium alloys. Eng Fail Anal 2004;11:715–25.

[16] Xue L. A unified expression for low cycle fatigue and extremely low cyclefatigue and its implication for monotonic loading. Int J Fatigue2008;30:1691–8.

[17] Usami T, Wang C-L, Funayama J. Developing high-performance aluminumalloy buckling-restrained braces based on series of low-cycle fatigue tests.Earthq Eng Struct Dyn 2012;41:643–61.

[18] Usami T. Developing high-performance damage control seismic dampers. In:10th symposium on ductile design method for bridges (Special Lecture): JSCE;2007. p. 11–22 [in Japanese].

[19] Kaufman JG. Properties of aluminum alloys: fatigue data and the effects oftemperature, product form, and processing: ASM Intl 2008.

[20] Usami T, Wang C-L, Funayama J. Improving low-cycle fatigue performance ofhigh-performance buckling-restrained braces by toe-finished method. J EarthqEng. http://dx.doi.org/10.1080/13632469.2012.703385.

[21] Stephens RI, Ali F, Stephens RR, Fuchs HO. Metal fatigue in engineering. 2nded. New York: John Wiley & Sons, Inc.; 2001.

[22] Tateishi K, Hanji T, Minami K. A prediction model for extremely low cyclefatigue strength of structural steel. Int J Fatigue 2007;29:887–96.