eqe.1103

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2011; 40:1553–1570Published online 8 March 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1103

Evaluation of plastic energy dissipation capacity of steel beamssuffering ductile fracture under various loading histories

Yu Jiao1,∗,†, Satoshi Yamada2, Shoichi Kishiki3 and Yuko Shimada4

1Department of Environmental Science and Technology, Tokyo Institute of Technology, J2-21, Nagatsuta 4259,

Yokohama 226-8503, Japan2Structural Engineering Research Center, Tokyo Institute of Technology, J2-21, Nagatsuta 4259, Yokohama

226-8503, Japan3Structural Engineering Research Center, Tokyo Institute of Technology, R3-12, Nagatsuta 4259, Yokohama

226-8503, Japan4Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

SUMMARY

The energy dissipation capacity of a structure is a very important index that indicates the structuralperformance in energy-based seismic design. This index depends greatly on the structural components thatform the whole system. Owing to the wide use of the strong-column weak-beam strength hierarchy wheresteel beams dissipate the majority of earthquake input energy to the structures, it is necessary to evaluatethe energy dissipation capacity of the beams. Under cyclic loadings such as seismic effects, the damage ofthe beams accumulates. Therefore, loading history is known to be the most pivotal factor influencing thedeformation capacity and energy dissipation capacity of the beams. Seismic loadings with significantlydifferent characteristics are applied to structural beams during different types of earthquakes and thereis no unique appropriate loading protocol that can represent all types of seismic loadings. This paperfocuses on the effects of various loading histories on the deformation capacity and energy dissipationcapacity of the beams. Cyclic loading tests of steel beams were performed. In addition, some experimentalresults from published tests were also collected to form a database. This database was used to evaluate theenergy dissipation capacity of steel beams suffering from ductile fracture under various loading histories.Copyright � 2011 John Wiley & Sons, Ltd.

Received 25 July 2010; Revised 13 December 2010; Accepted 16 December 2010

KEY WORDS: steel beam; cyclic loading test; energy dissipation capacity; loading history; ductile fracture

1. INTRODUCTION

The balance between seismic input energy and the energy dissipation capacity of the structures isthe principal concept of energy-based seismic design [1]. A structure’s energy dissipation capacity,which depends greatly on its structural components, is an important index of earthquake resistance.Steel moment frames with strong-column weak-beam mechanism are frequently used in earthquake-prone areas. In such a system, plastic hinges form in the beams and dissipate the majority of theearthquake input energy, making the beams the dominant anti-earthquake components in the wholestructure.

Under cyclic loadings such as earthquake effects, the damage of the beams accumulates. Thus,loading history is known as the most definitive factor that affects the deformation capacity and

∗Correspondence to: Yu Jiao, Department of Environmental Science and Technology, Tokyo Institute of Technology,J2-21, Nagatsuta 4259, Yokohama 226-8503, Japan.

†E-mail: [email protected], [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

1554 Y. JIAO ET AL.

0

− Δy

− 2 Δy

− 3 Δy

− 4 Δy

2 Δy

3 Δy

4 Δy

Δy

ATC-24

Def

orm

atio

n

0

Dri

ft a

ngle

(ra

d.)

− 0.01

− 0.02

− 0.03− 0.04

− 0.05

0.01

0.02

0.03

0.04

0.05

SAC

− 1.0

00.048

− 1.0

Rel

ativ

e am

plit

ude

(ai/

Δm)

FEMA 461

4 θp

− 4 θp

8 θp

− 8 θp

0

2 θp

6 θp

− 2 θp

− 6 θp

Japan Rec.

Rot

atio

n

Figure 1. Recommended standard loading protocols.

energy dissipation capability of the beams. Seismic effects of significantly different characteristicsare applied to structures during distinct earthquakes such as near-fault earthquakes or offshoreinterplate earthquakes. Furthermore, the number and amplitudes of the loading cycles the beamexperiences during earthquakes depend on the configuration, strength, stiffness, etc. of the structure.

Figure 1 shows some of the most common recommended standard loading protocols employedin steel beam cyclic loading tests including the ATC-24 protocol [2], the SAC protocol [3], andthe FEMA 461 [4] protocol, which are widely used in the U.S.A., and the loading protocolrecommended by the Building Research Institute and the Japan Iron and Steel Federation [5],which is commonly accepted in Japan. Among the U.S.A. protocols, the ATC-24 is similar tothe Japanese one. The SAC protocol includes more small elastic cycles because of the observedNorthridge weld fractures that occurred before yielding occurred [6]. However, some researchersand engineers are more concerned about the cumulative plastic deformation capacity of the beamafter yielding. The advantage of introducing the recommended loading protocols is that it is easy tocompare the performances of different specimens under the same loading protocol. A majority of theloading protocols are incremental deformation amplitudes loadings, which seem to be inadequateto represent different earthquake loadings. It is also indicated that a comprehensive testing programfor structural components should include a monotonic test in addition to cyclic tests [7]. Thereis no unique best loading history that can represent all kinds of seismic loadings applied to thebeams. In order to assess the consequences of cumulative damage, more loading histories shouldbe employed in addition to the standard loading protocols in the steel beam tests. Under suchcircumstances, it is necessary to seek a concise method of investigating the seismic performancesof beams with different structural details that are subjected to various loading histories.

Ductile fracture is one of the typical ultimate states of steel beams in earthquakes. Duringthe 1995 Kobe earthquake, a great deal of beam damage due to brittle fracture was observed.After that, according to the new Japanese design code, the most dangerous brittle fracture shouldbe avoided. In addition, the current material and welding methods also help to prevent brittlefracture. Therefore, this paper focuses on beams that suffer ductile fractures at the flanges. Theseismic performance of the beams is discussed in terms of energy. Five large-scale beam–column

Copyright � 2011 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2011; 40:1553–1570DOI: 10.1002/eqe

EVALUATION OF PLASTIC ENERGY DISSIPATION CAPACITY 1555

R35

Backing plate

(FB-9 25)x

Backing plate

(FB-9 25)x25

735

35

R10

Spot weldingH-400 200 8 13(SN400B)

RHS-400 400 19(BCR295)

1900

12-PL9-374 96

Detail

x x

x xx

x

PL-450 450 16x x

Diaphragm

Figure 2. Specimen details.

subassemblies were tested under different loading histories with incremental, decremental, andconstant amplitudes in order to understand the influence of the diversity of the loading historieson the beams. In addition, 24 specimens from six published steel beam–column subassembly tests[8–13] were also investigated. Based on the complete experimental database, the energy dissipationcapacity of the steel beams with the failure mode of ductile fracture under various cyclic loadinghistories was evaluated.

2. CYCLIC LOADING TESTS OF STEEL BEAMS

2.1. Objective

Cyclic loading tests of five T-type beam–column subassemblies (Nos. 1−5 in the database) withexactly the same structural details were conducted to investigate the effect of the loading history onthe energy dissipation capacity of steel beams. The only variable in this experiment is the loadinghistory.

2.2. Specimens

The column of each specimen was a rectangular hollow section (RHS) 400mm×400mm×19mm(BCR295 steel) while the beam was a wide-flange section 400mm (depth)×200mm (width)×8mm (web thickness) ×13mm (flange thickness) (SN400B steel) (Figure 2). The beam was shopwelded onto the column via through-diaphragms with the improved type of weld access holes andsolid end tab in accordance with the Japanese Standard Specification JASS 6 [14]. The steel platesof the column and panel zone were thick enough so that neither yielded even when the beamreached its maximum strength. Furthermore, in order to prevent local buckling, stiffeners werewelded to the beams near the connections as well as the loading points. Tensile coupon tests ofboth the flange and web were performed using JIS-1A testing samples [15]. The results are shownin Figure 3.

2.3. Test setup

Each specimen, which was rotated 90◦ with its beam standing vertically and column lying horizon-tally, was setup through column jigs and screw jacks, with the former fastened onto the reactionframe for vertical support and the latter contacting the reaction frame for horizontal support(Figure 4). A loading jig connecting the free end of the beam and an oil jack, which was installedhorizontally on the reaction frame, made up the loading system. In addition, lateral supports (stiff-ening systems) were set at three locations to prevent lateral buckling of the beam. The beamrotation is defined in Equation (1) and illustrated in Figure 5.

�= �

L−� (1)


1556 Y. JIAO ET AL.

0

100

200

300

400

500

600

0

FlangeWeb

Nom

inal

str

ess

σ [N

/mm

2 ]

Nominal strain ε [%]

Yielding Point Strength Elongation

N/mm EL

Flange 282 452 32

Web 352 514 30

5 10 15 20 25 30 35

Figure 3. Coupon test results and specimen mechanical characteristics.

Oil Jack Lateral Support

Reaction Frame

Specimen

Screw Jack Column Jig

Loading Jig

Figure 4. Setup of cyclic loading tests.

Specimen

Column's Face line

Column's Face line Specimen

Figure 5. Definition of beam rotation.



− 4 θp

− 8 θp

− 2 θp

− 6 θp

Loading 1

4 θp

8 θp

0

2 θp

6 θp

Loading 2

4 θp

8 θp

0

2 θp

6 θp

− 4 θp

− 8 θp

− 2 θp

− 6 θp

4 θp

8 θp

0

2 θp

6 θp

Loading 3

− 4 θp

− 8 θp

− 2 θp

− 6 θp

4 θp

8 θp

0

2 θp

6 θp

Loading 4

− 4 θp

− 8 θp

− 2 θp

− 6 θp

4 θp

8 θp

0

2 θp

6 θp

Loading 5

− 4 θp

− 8 θp

− 2 θp

− 6 θp

Figure 6. Loading histories.

2.4. Loading histories

Five different deformation-controlled cyclic loadings, shown in Figure 6, were applied to thespecimens during the test with the unit beam rotation set to �P (0.0058 rad by calculation), theelastic beam rotation when the moment of the beam’s cantilever end equals its full plastic momentMP (383kNm by calculation). The five loading histories are as follows Loading1: Incrementalcyclic loading starting from ±2�P , with intervals of 2�P . For each loading step, two cycles wereapplied. Loading 2: Decremental cyclic loading completely converse of loading (1), i.e. loadingstarting from the amplitude where the specimen subjected to loading (1) fractured, with intervalsof −2�P . For each loading step, two cycles were applied. Loadings 3–5:: Cyclic loading with aconstant amplitude of ±3�P , ±4�P , and ±5�P , respectively. Loading (1) is recommended by theBuilding Research Institute and the Japan Iron and Steel Federation [5]. Considering the seismiceffects of small amplitude shaking after large ones, the decremental loading (2) was also includedin the loading histories. Loadings (3)–(5) were employed to simulate earthquake loadings underlong-duration ground motion.

2.5. Experimental results and discussion

The load–deformation relationships obtained from the cyclic loading tests (hysteresis loops) areshown in Figure 7. Ductile cracks on the beam flange were first observed near the toe of theweld access hole of each specimen, which propagated longer and wider during the test until thespecimen reached fracture. Here, fracture is defined as the point when the measured load starts todecrease while the specimen’s deformation is still increasing. The processes of the cyclic loadingtests were as follows. Loading 1: Loading started from two cycles of ±2�P , until two cycles of±6�P , as scheduled. After that, the specimen fractured during the first loading cycle of +8�P . Thefracture point was confirmed when the cumulative rotation was approximately 0.605 rad. Loading 2:Loading started with a half cycle of +8�P , which was the amplitude where the specimen underloading (1) fractured, followed by the decremental cyclic loading mentioned above (from −6�Pto +2�P ). The specimen did not fracture until it was loaded again toward +8�P . The fracturepoint was confirmed at the cumulative rotation of 0.677 rad. Loading 3: 26 cycles of loading witha constant amplitude of ±3�P were completed. The specimen fractured as it was approaching+3�P of the 27th cycle, when the cumulative rotation equaled 1.863 rad. Loading 4: 9.5 cycles ofloading with a constant amplitude of ±4�P were completed. The specimen fractured on its way to−4�P in the 10th cycle, when the cumulative rotation equaled 0.935 rad. Loading 5: 3 cycles of


1558 Y. JIAO ET AL.

-800

0

800

-0.05 0 0.05

Loading 1

Mom

ent (

kN-m

)

Beam rotation θ (rad)

2θ 4θ 6θ-2θ-4θ-6θ

-800

0

800

-0.05 0 0.05

Loading 2

Mom

ent (

kN-m

)


2θ 4θ 6θ-2θ-4θ-6θ 8θ

-800

0

800

-0.05 0 0.05

Loading 3

Mom

ent (

kN-m

)


3θ-3θ

-800

0

800

-0.05 0 0.05

Loading 4

Mom

ent (

kN-m

)


4θ-4θ

-800

0

800

-0.05 0 0.05

Loading 5

Mom

ent (

kN-m

)


5θ-5θ

Figure 7. Loading-deformation relationships.

Skeleton Curve

Elastic Unloading Part

Bauschinger Part θ

M M

00

Figure 8. Decomposition of beams’ hysteresis loops.

loading with a constant amplitude of ±5�P were completed. The specimen fractured on its wayto +5�P in the 4th cycle, when the cumulative rotation equaled 0.400 rad.

Load–deformation hysteresis loops of steel beams under cyclic loading can be decomposedinto three parts: the skeleton curve, Bauschinger part, and elastic unloading part [1], as illustratedin Figure 8. Here, the skeleton curve is obtained by connecting parts of the load–deformationrelationship sequentially when the beam first experienced its maximum load (both positive andnegative). It is observed that the load–deformation relationships of the steel beams under monotonicloading have an approximate correspondence to the skeleton curves of the steel beams withsame structural details that fracture within relatively fewer loading cycles under cyclic loadings.Moreover, assuming the Bauschinger part to be elastic and neglecting the Bauschinger part donot result in unsafe-side estimation of energy absorption capacity, although the estimate mightbe too conservative [1]. Therefore, there is an empirical rule that the deformation capacity andenergy dissipation capacity of the skeleton curves are adopted to characterize the behavior of thebeams [1]. Nevertheless, this empirical rule was obtained under the condition that the steel beamsfracture within relatively few loading cycles under cyclic loadings. Under certain loading historiessuch as long-duration ground motions, a beam fractures after many loading cycles. In these cases,the skeleton curves obtained are obviously shorter than the load–deformation relationships of thebeams under monotonic loading. The energy dissipation capacity in the Bauschinger parts becomestoo large to be neglected, rendering this empirical rule no longer applicable. In this paper, theexperimental results were also studied by first dividing the hysteresis loops into skeleton curves andBauschinger parts similar to the abovementioned methodology [1], while taking into considerationthe contributions of the Bauschinger parts as well in the second step, which was neglected in [1].



-800

0

800

0 0.02 0.04 0.06 0.08

Loading 1

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.02 0.04 0.06 0.08

Loading 2

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.02 0.04 0.06 0.08

Loading 3

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.02 0.04 0.06 0.08

Loading 4

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.02 0.04 0.06 0.08

Loading 5

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

Figure 9. Skeleton curves.

-800

0

800

0 0.125 0.25 0.375 0.5

Loading 1

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.125 0.25 0.375 0.5

Loading 2

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.125 0.25 0.375 0.5

Loading 3

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.125 0.25 0.375 0.5

Loading 4

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

-800

0

800

0 0.125 0.25 0.375 0.5

Loading 5

PlusMinus

Mom

ent (

kN-m

)

|θ | (rad)

Figure 10. Bauschinger parts.

The skeleton curves and Bauschinger parts of both the plus and minus loading sides derivedfrom the load–deformation relationships are shown in Figures 9 and 10. Very similar skeletoncurves can be observed from loadings (1), (2), and (5), which share the same characteristics ofthe comparatively large maximum deformation amplitudes; the specimens under these loadinghistories reached ductile fracture points within a fewer number of loading cycles. On the otherhand, the skeleton curves of loadings (3) and (4) with relatively small deformation amplitudes andlarge numbers of loading cycles are considerably shorter than are those of the others. Althoughthe skeleton curves of all specimens have similar shapes, they become shorter when the maximumloading amplitudes decrease, whereas the Bauschinger parts grow remarkably larger. The decreasein the energy dissipation capacity in the skeleton curves is much less than is the increase inthe energy dissipation capacity in the Bauschinger parts. Thus, the ‘sacrifice’ of the skeletoncurves results in larger overall energy dissipation capacity. In order to evaluate the energy dissi-pation capacity of the beams under various loading histories, it is necessary to clarify the rela-tionship between the energy dissipation capacity of the skeleton curves and the Bauschingerparts.


1560 Y. JIAO ET AL.

3. RELEVANT EXPERIMENTS OF THE STEEL BEAMS

Even though some basic information of the effects of loading histories on the energy dissipationcapacity of steel beams was obtained from the experiment mentioned in Section 2, the number anddiversity of the specimens are rather limited. To form a stronger argument, additional publishedexperimental data were included in the database for evaluating the energy dissipation capacity ofthe beams. The load–deformation results from six published beam-to-column subassembly (shop-welded flange and web connection) experiments were investigated. Stiffeners were welded near theconnections on all specimens (beam parts) to prevent local buckling. The ultimate states of all ofthese specimens were confirmed to be ductile fractures at the beam flanges near the connections.Here, the definition of fracture is the same as that illustrated in the previous cyclic loading tests.Together with the cyclic loading tests mentioned in Section 2, details of the database are listed inTable I. For the wide-flange section, H is the beam depth, B is the beam width, and tw and t f arethe web and flange thicknesses, respectively. For the rectangular hollow section, D is the columnwidth and t is the thickness.

Nos. 6–9 are specimens from the monotonic loading tests [8]. The shape of the column cross-section (RHS column/wide-flange column), the thickness of the column, and the presence ofthe weld access holes are the variables of this experiment. The cross-sections and spans of thewide-flange beams were kept constant.

Seven wide-flange beam-to-RHS-column connection specimens, numbered 10−16, from afull-scale shaking table test [9] were also collected (Table I). The NS component of the JMAKobe record (according to the Japan Meteorological Agency of Kobe, 1995), which was scaledto the peak velocity of 1.0 m/s, was input during testing. Note that due to the characteristicsof the input record (near-fault earthquake record), each specimen fractured within few plasticloading cycles, meaning the skeleton curves obtained from this experiment would be verysimilar to the load–deformation relationships of the beams if they were subjected to monotonicloading [1].

Thirteen specimens (Nos. 17−29) from four cyclic loading tests [10–13] are also includedin the database (Table I). Eight of them are steel beam specimens with 50 mm thick end plates(PL in Table I); the others are beam–column subassemblies with wide-flange beams and RHScolumns.

4. ENERGY DISSIPATION CAPACITY OF STEEL BEAMS UNDER RANDOM LOADINGHISTORIES

4.1. Equivalent cumulative plastic deformation ratio

The energy dissipation capacity of steel beams was expressed by means of the Equivalent Cumula-tive Plastic Deformation Ratio (�), which is one of the indices used to quantify cumulative damagein energy-based seismic design. Equation (2) shows the definition of �:

�= Wp

Mp ·�p(2)

where Wp is the plastic energy dissipation capacity of a steel beam, Mp is the full plasticmoment of the steel beam, and �p is the beam rotation when the moment at the beam endreaches Mp.

The equivalent cumulative plastic deformation ratio of steel beams subjected to monotonicloading �0 is illustrated in Figure 11, which is defined as

�0 = Wp0

Mp ·�p(3)

where Wp0 is the plastic energy dissipation capacity of a steel beam under monotonic loading.



Tabl

eI.

Dat

abas

ede

tails

.

Bea

mC

onne

ctio

nC

olum

n

Cro

ssse

ctio

nSp

anFl

ange

Web

Wel

dac

cess

Cro

ssse

ctio

n�

yL

oadi

ngN

umbe

rof

Num

ber

W−

H×

B×t

w×t

f(m

m)

�y(N

/m

m)2

YR

(%)

�y(N

/m

m)2

YR

(%)

Hol

e(m

m)

�−

D×t

(N/m

m2)

hist

ory

cycl

es�

max

/�

p

1W

−400

×200

×8×1

321

0028

262

352

6835

�−4

00×1

932

4.5

Cyc

lic6.

56.

552

W−4

00×2

00×8

×13

2100

282

6235

268

35�

−400

×19

324.

5C

yclic

78

3W

−400

×200

×8×1

321

0028

262

352

6835

�−4

00×1

932

4.5

Cyc

lic26

.53

4W

−400

×200

×8×1

321

0028

262

352

6835

�−4

00×1

932

4.5

Cyc

lic10

45

W−4

00×2

00×8

×13

2100

282

6235

268

35�

−400

×19

324.

5C

yclic

3.5

56

[8]

W−3

00×1

50×6

.5×9

1500

334

7037

576

0W

−250

×250

×10×1

434

8M

ono

0.5

17.7

67

[8]

W−3

00×1

50×6

.5×9

1500

334

7037

576

0�

−250

×936

4M

ono

0.5

15.2

88

[8]

W−3

00×1

50×6

.5×9

1500

334

7037

576

25�

−250

×936

4M

ono

0.5

9.85

9[8

]W

−300

×150

×65×9

1500

334

7037

576

25�

−250

×631

2M

ono

0.5

9.25

10[9

]W

−600

×300

×12×2

527

2933

064

413

7635

�−5

00×1

650

1Sh

akin

g2

12.3

511

[9]

W−6

00×3

00×1

2×2

527

2933

064

413

7635

�−5

00×2

249

8Sh

akin

g3

13.6

512

[9]

W−6

00×3

00×1

2×2

527

2933

064

413

7635

�−5

00×2

249

8Sh

akin

g3

13.9

113

[9]

W−6

00×3

00×1

2×2

527

2933

064

413

7635

�−5

00×2

249

8Sh

akin

g3

14.1

14[9

]W

−600

×300

×12×

2527

2933

064

413

760

�−5

00×2

234

2Sh

akin

g14

12.9

415

[9]

W−6

00×3

00×1

2×2

527

2933

064

413

7635

�−5

00×3

240

5Sh

akin

g14

.513

.92

16[9

]W

−600

×300

×16×3

227

2931

662

378

7035

�−5

00×3

240

5Sh

akin

g2.

515

.28

17[1

0]W

−600

×200

×9×1

220

5029

168

385

7435

�−4

00×1

938

8In

cr.

34.

7218

[11]

W−6

00×2

00×1

1×1

737

2529

962

334

680

PL−5

0In

cr.

86

19[1

1]W

−600

×200

×11×1

737

2529

962

334

680

PL−5

0In

cr.

17.5

620

[11]

W−6

00×2

00×1

1×1

737

2529

962

334

680

PL−5

0C

on.

11.5

421

[11]

W−6

00×2

00×1

1×1

737

2529

962

334

680

PL−5

0C

on.

4.5

622

[11]

W−6

00×2

00×1

1×1

722

2529

962

334

680

PL−5

0In

cr.

8.5

623

[11]

W−6

00×2

00×1

1×1

722

2529

962

334

680

PL−5

0In

cr.

18.5

624

[11]

W−6

00×2

00×1

1×1

722

2529

962

334

680

PL−5

0C

on.

8.5

425

[11]

W−6

00×2

00×1

1×1

722

2529

962

334

680

PL−5

0C

on.

76

26[1

2]W

−400

×200

×8×1

319

0031

471

368

7835

�−4

00×1

935

9In

cr.

6.5

6.1

27[1

2]W

−400

×200

×8×1

319

0031

471

368

7835

�−4

00×1

935

9In

cr.

117.

628

[13]

W−4

88×3

00×1

1×1

821

5038

673

396

7135

�−4

00×1

939

7In

cr.

106

29[1

3]W

−488

×300

×11×1

821

5031

872

318

7235

�−4

00×1

926

7In

cr.

10.5

6


1562 Y. JIAO ET AL.

Q θM

θ

M

0

Mp

θp

P0W

Figure 11. Plastic energy dissipation of steel beam under monotonic loading.

θ

Bauschinger Part

θ

M M

00 θ

Skeleton CurveM

0

WS

WB

Figure 12. Decomposition of the hysteresis loop and plastic energy dissipation of steelbeams under cyclic loading.

In case of beams under cyclic loadings, with the decomposition of the hysteresis loops, theplastic energy dissipation capacity of a steel beam Wp is expressed as the sum of the plasticenergy dissipation capacity from both their skeleton curves and Bauschinger parts, as shown inEquation (4) and Figure 12:

Wp =WS +WB (4)

where WS and WB are the plastic energy dissipation capacity of the skeleton curve and Bauschingerpart, respectively. Consequently, it is possible to express the equivalent cumulative plastic defor-mation ratio of steel beams under cyclic loading � in Equations (5)–(7).

� = S�+ B� (5)

S� = WS/(Mp ·�p) (6)

B� = WB/(Mp ·�p) (7)

where S� and B� are the equivalent cumulative plastic deformation ratio of the skeleton curve andBauschinger part, respectively. In this study, the equivalent cumulative plastic deformation ratio ofeach part of the hysteresis loop was studied to investigate the energy dissipation capacity of steelbeams subjected to cyclic loading.

4.2. Energy dissipation capacity of the steel beams under monotonic loading

4.2.1. Outline. The loading history is one of the principal factors that affect the plastic deforma-tion capacity of steel beams. However, in order to evaluate a beam’s energy dissipation capacity,the mechanical characteristics of the material and its geometric characteristics must also be takeninto consideration. Among them, the yield point and yield ratio of the steel together with thebeam’s section and span affect the full plastic moment, initial stiffness, and the post-yield stiff-ness of the beam. The size of the weld access holes as well as the column cross-section greatly



(Weld access hole)

(Rlocal bending of the column tube wall

HS column)

Loss of section

Figure 13. Local bending of column tube wall and loss of web section.

affect the joint efficiency of the beam web. Specifically, the weld access holes (loss of section)cause a decrease in a beam’s moment of plastification. The tube walls of a column (especiallyRHS column) at the connections tend to have an out-of-plane deformation when the beams aresubjected to bending. These are the main reasons for the inefficiency of the connection’s momenttransmission [8], i.e. the joint efficiency of the beam web is less than 100% (Figure 13). Theinfluence of the material and geometric characteristics can be assessed by investigating beamsunder monotonic loading. Then, the energy dissipation capacity of beams under cyclic loadinghistories is studied based on the energy dissipation capacity of the beams under monotonicloading.

According to the current research, the concept of a bilinear model of a beam’s load–deformationrelationship under monotonic loading was introduced to illustrate the effect of joint efficiency.The relationship between the strain concentration ratio of the beam flange and the ultimate beammoment was obtained [8]. Nevertheless, there is still a lack of quantitative studies of deformationcapacity/energy dissipation capacity. In this study, a quantitative bilinear model was developed toobtain the monotonic plastic deformation capacity of a beam.

4.2.2. Parametric study of a beam’s load–deformation relationship. In order to obtain a reasonablesecond stiffness ratio of the bilinear load–deformation model, a parametric study of the materialcharacteristics and the beam moment gradient was conducted through an in-plane analysis [16–18]of the ideal cantilever wide-flange beams subjected to monotonic shear bending (without out-of-plane deformation). The cross-section of the analytical beam model was set to a wide-flangesection of 600mm (depth)×300mm (width)×13mm (web thickness)×25mm (flange thickness).In order to change the beam moment gradient, three different beam spans were defined: 3, 4,and 6 m. In addition, four stress–strain relationship models (material characteristics) with differentyield stresses/strains and yield ratios were introduced in this analysis. Two types of steel withdifferent yield points and elongations that are commonly used in Japan as steel beams, SN400and SN490, were chosen as typical materials. The tensile coupon test results of these two typesof steel were used as the basic stress–strain models. The experimental yield ratio of SN400 wasaround 60% while that of SN490 was around 70%. It is possible to obtain two more relevantstress–strain models with different yet reasonable yield ratios (70% for SN400 and 80% for SN490)by varying the post-yield stress value. The input stress–strain relation models and the outputanalytic results are shown in Figures 14 and 15. There is hardly any significant difference betweenthe second stiffness ratios of the load–deformation relationships, despite the different parameters.Therefore, it is feasible to assume that the second stiffness ratio of the ideal steel wide-flange beamload–deformation relationships under monotonic loading is a constant, which is considered to bearound 2.5%.

4.2.3. Web joint efficiency at the ultimate state. It is necessary to discuss the joint efficiency of thebeam web at the beam-to-column connection at its ultimate state in order to ascertain the ultimate


1564 Y. JIAO ET AL.

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25

SN400-60%SN400-70%SN490-70%SN490-80%N

omin

al s

tres

s (N

/mm

2 )

Nominal strain

Figure 14. Input stress–strain relationship models.

0

0.5

1

1.5

2

0 10 20 30

θ /θp

M /

Mp

Bilinear modelAnalytic results

Figure 15. Results of the parametric study.

load and rotation of the beam under monotonic loading. Ultimate joint efficiency �w is defined as

�w = j Mwu

Mwu(8)

where j Mwu is the maximum moment of the beam web at the connection considering the loss ofthe beam section and local bending of the column tube wall, Mwu =�wu · Zwp is the ideal maximummoment of the beam web at the connection, �wu is the ultimate stress of the beam web, and Zwpis the plastic section modulus of the full section beam web. Here, j Mwu is calculated by modifiedEquations (9) and (10) based on the Japanese ‘Recommendation for design of connections in steelstructures’ [19].

j Mwu = m · Zwpe ·�wy (beams with weld access holes) (9)

j Mwu = m · Zwpe ·�wu (beams without weld access holes) (10)

where �wy is the yield stress of the beam web, Zwpe = 14 (Db −2tbf −2Sr )2 ·tbw is the plastic section

modulus of the beam web considering the loss of cross-section, and m [19, 20] is the normalizedbending strength of the beam web at the beam-to-column connection m = j Mwu/Mwp, where Mwpis the full plastic moment of the beam web considering the loss of section due to the weldaccess holes. Here, m is controlled by the size of the column, beam and the weld access holesas well as the yield strengths of the column and beam. In design, it is possible to approximatem as follows: for the wide-flange columns (strong-axis direction), m =1; for the RHS columns,m =min{1, 4(tcf /d j )

√(b j ·�cy)/(tbw ·�wy)}, where �cy is the yield stress of the column, Db is the

beam height, tbf is the thickness of the beam flange, Sr is the loss of cross-section in the beamheight dimension (usually the length of the weld access hole along the beam height), tbw is the



0

20

40

60

80

100

120

0 20 40 60 80 100 120

Modi. Meth.

AIJ Rec. Meth.

Cal

cula

ted

valu

e (%

)

Experimental value (%)

Figure 16. Verification of the joint efficiency calculation method.

thickness of the beam web, tcf is the steel plate thickness of the RHS column, d j is the innerdistance of two diaphragms, b j = Bc −2tcf is the width of the yield area on the RHS columnobtained from the yield line theory, and Bc is the width of the RHS column.

In [19], only Equation (9), which originates from plastic analysis where the material is consid-ered to be elastic-perfectly plastic, is recommended. Nevertheless, strain hardening occurs in realstructural steel. In the wide-flange beams without weld access holes, the web works more effi-ciently and reaches the maximum stress �wu at the ultimate state. Equation (10) is introduced inthis study for beams without weld access holes.

The experimental results of the 11 specimens (specimen Nos. 6−9 from the monotonic loadingtests [8] and specimen Nos. 10−16 from the shaking table tests [9]) from the database wereinvestigated to verify the proposed evaluation method of joint efficiency. Note that although thespecimens from the shaking table tests were subjected to cyclic loadings, under the JMA Koberecord, the specimens reached the maximum load within very few loading cycles. The straindata of the measuring steps in the load–deformation skeleton curve were connected, which couldapproximately be regarded as the strain history under monotonic loading. The calculated jointefficiencies of these 11 specimens through the recommended [19] as well as the modified evaluationmethod were compared with the experimental values calculated by Okada et al. [8]. Verification ofthe evaluation methods is shown in Figure 16 with the experimental results plotted on the x-axisand the calculated results plotted on the y-axis. The modified method shows higher accuracy.Particularly for specimens without weld access holes, the results of the modified method areconsiderably closer to the experimental results (arrows in Figure 16).

4.2.4. Bilinear load–deformation model. A bilinear load–deformation relationship model isproposed here based on the model mentioned in [8] to evaluate the energy dissipation capacity ofsteel beams under monotonic loading. Figure 17 shows details of the model. Following the previousdiscussion, it is necessary to divide the beams into two categories: with and without weld accessholes. For both categories, K is the initial stiffness and Kst =2.5%·K is the second stiffness.Mu and �u are the actual ultimate load–deformation conditions considering the decrease in jointefficiency. For beams without weld access holes, the plastification moment can be represented byMp and �p. Mui and �ui are the ideal ultimate moment and beam rotation, respectively, when thejoint efficiency �w is 100%, which can be obtained from the model with some basic structuralinformation. For beams with weld access holes, the loss of the beam section causes a decreasein the plastification moment. Therefore, it is necessary to calculate the full plastic moment fromthe net beam section. Thus, the plastification moment M ′

p and the corresponding beam rotation�′

p are smaller than the commonly defined values of Mp and �p. Mui is the same as the ideal


1566 Y. JIAO ET AL.

KK st ⋅= %5.2Actual ultimate condition(Joint efficiency<100%)

Ideal ultimate condition(Joint efficiency=100%)

M

MuiMu

Mp

p u ui

Kst

W

K

M

Mui

p ui

K

Mp Kst

Mu

u

W

Mp

p

,

,

Steel beams with weld access holes Steel beams without weld access hole

Ideal ultimate condition(Joint efficiency=100%)

Actual ultimate condition(Joint efficiency<100%)

,θθθ θ θθ θ θθ

Figure 17. Bilinear load-deformation model of steel beams under monotonic loading.

0

5

10

15

20

25

0 5 10 15 20 25

η0 (Exp.)

η 0 (

Cal

.)

1

23

4

Figure 18. Verification of �0 of steel beams under monotonic loading.

ultimate fracture moment of the beams without weld access holes while �′ui is the corresponding

beam rotation. The actual maximum moment of the beam Mu is expressed in Equation (11) as

Mu = Mfu +�w · Mwu (�w�1) (11)

where Mfu =�fu · Zfp is the maximum moment of the beam flange at the connection and Zfp isthe plastic section modulus of the full section beam flange. From the joint efficiency �w, it ispossible to obtain the maximum beam rotation �u from the bilinear model. In other words, theactual ultimate load–deformation condition of a steel beam subjected to monotonic loading can bederived.

Therefore, through this model, it is possible to calculate the energy dissipation capacity Wp0as well as the equivalent cumulative plastic deformation ratio �0 of steel beams subjected tomonotonic loading. The experimental results from the monotonic loading tests [8] were investigatedto verify the evaluation method. Figure 18 shows the verification. The experimental results of�0 from the monotonic loading tests are plotted on the x-axis and the calculated results of �0are plotted on the y-axis. The graph shows good correspondence between the experimental andcalculated values of the monotonic energy dissipation capacity of the beams. The effects of themechanical characteristics of the material and the geometric characteristics of the beam on theenergy dissipation capacity of the beam were obtained.

4.3. Energy dissipation capacity of steel beams under various cyclic loadings

4.3.1. Normalization of the equivalent cumulative plastic deformation ratio. In order to removethe influence of the material and geometric characteristics on the energy dissipation capacity ofsteel beams, the equivalent cumulative plastic deformation ratios of the skeleton curves (S�) andBauschinger parts (B�) of the beams as well as the overall equivalent cumulative plastic deformationratios �, were normalized by the equivalent cumulative plastic deformation ratios of steel beamsunder monotonic loading �0. The normalized values S�/�0, B�/�0, and �/�0 can be regarded as the



0

0.5

1

1.5

2

2.5

3

0 5 10 15 20θmax/θp

Sη/η

0

θθηη /15.0/ ⋅=

θ

M θ

Figure 19. Relationship between the normalized equivalent cumulative plastic deformation ratio of theskeleton curve and the normalized maximum deformation.

energy dissipation capacity of the steel beam with the same material and geometric characteristics,which is only affected by the different loading histories.

4.3.2. Maximum deformation and the energy dissipation capacity. This section discusses the rela-tionship of the energy dissipation capacity of a beam and the maximum beam deformation duringthe loading procedure. The maximum deformation �max is defined as the absolute value of themaximum beam rotation experienced by the specimens under cyclic loadings, which is normalizedby �p (Table I) for comparison between different specimens. The relationship between the normal-ized skeleton equivalent cumulative plastic deformation ratio S�/�0 and the normalized maximumdeformation �max/�p of the database is plotted in Figure 19. The plots show an approximatelylinear relationship, which is expressed in Equation as (12),

S�/�0 =0.15 ·�max/�p (3��max/�p�17.76) (12)

4.3.3. Number of loading cycles and energy dissipation capacity. This section looks at the energydissipation capacity of the steel beams under various loading histories by investigating its relation-ship with the number of loading cycles the beams survive until ductile fracture. For the previouslymentioned shaking table tests, it is necessary to count the number of effective loading cyclesuntil fracture, i.e. the loading cycles with their amplitudes in the plastic region that contribute toplastic energy dissipation. The rainflow-counting algorithm originally published in [21] was usedto calculate the number of loading cycles in the shaking table tests. This algorithm defines cycles asclosed load/deformation hysteresis loops. According to the load–deformation relationship recordedduring the tests, a large number of loading cycles experienced by the specimen was within theelastic region, which contributed little to the total plastic energy dissipation. Therefore, cycles withloading amplitudes less than 2/3 Mp, which hardly influenced the energy dissipation capacity ofthe specimens, were skipped in the cycle counting.

In the database (Table I), the number of loading cycles until the beam fractures, N , varies from0.5 to 26.5. The relationships of the normalized equivalent cumulative plastic deformation ratiosof the specimens and the number of loading cycles until the specimens fracture are plotted inFigure 20. S�/�0 decreases with an increase in N , while B�/�0 and �/�0 show opposite trends.The fitted line of each relationship is also plotted in these graphs It is possible to approximatethe normalized equivalent plastic deformation ratios of the specimens as functions of ln(N n) withtolerable errors, as illustrated in Equations (13)–(15).

S�/�0 = ln N 5/4 +1.9 (0.5�N�1)

S�/�0 = − ln N 5/12 +1.9 (1<N�26.5)(13)


1568 Y. JIAO ET AL.

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Bauschinger Part

Bη/

η 0

Number of Loading Cycles

ln/ N=ηη

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Skeleton Curve

Sη/η 0


9.1ln/ +−= Nηη

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

Overall

η/η 0


9.1ln/ += Nηη

Figure 20. Relationship between the normalized equivalent cumulative plastic deformation ratio of theskeleton curve and the number of loading cycles until fracture.

B�/�0 = 0 (0.5�N�1)

B�/�0 = ln N 5/3 (1<N�26.5)(14)

�/�0 = ln N 5/4 +1.9 (0.5�N�26.5) (15)

Note that, different from the commonly used Miner’s rule, proposed Equations (12)–(15) arebased on the decomposition of the beams’ hysteresis loops, which can only be applied after theplastification of the steel beams subjected to relatively large amplitude cyclic loadings. SinceEquations (12)–(15) were obtained from the experimental results, the applicable ranges of theseequations are considered to be within the scope of the database, ie. (0.5�N�26.5). Consideringthe characteristics of the seismic loadings applied on steel beams, the amplitudes are rather largeand the total numbers of shaking cycles, especially those that cause damage to structures, is quitelimited, usually a few tens. Therefore, Equations (12)–(15) can be considered valid in seismicdesign However, when it comes to the loading cycles whose amplitudes are within the elastic regionwhere the number of loading cycles is extremely large, these equations are no longer applicable.

5. CONCLUSION

The energy dissipation capacity of the structural components plays a significant role in energy-based seismic design because it is the index of structural performance. Considering the diversityof earthquake loadings applied to steel beams, this study aims at evaluating the energy dissipationcapacity of beams ending in ductile fracture under various loading histories.

Five beam–column subassemblies were tested under different cyclic loadings to study the effectsof various loading histories on the energy dissipation capacity of steel beams. Together with somepublished experimental results, a database of steel beams tested under diverse loading protocolswas created. The energy dissipation capacity of the steel beams under various loading historieswas evaluated by studying the energy dissipation capacity of the skeleton curves and Bauschingerparts of the specimens in the database. The conclusions of this study are summarized as follows:



(1) Under various loadings, a linear relationship was found between the normalized equivalentcumulative plastic deformation ratio of the skeleton curve (S�/�0) and the normalizedmaximum deformation (�max/�p).

(2) The normalized equivalent plastic deformation ratio of each part of the beam’s load-deformation relationship (S�/�0 and B�/�0) and the overall normalized equivalent plasticdeformation ratio (�/�0) can be approximately expressed by the natural logarithmic func-tions of the number of loading cycles that the beam experiences in its plastic region untilfracture (N ).

(3) It is possible to evaluate the energy dissipation capacity of steel beams that suffer ductilefracture under random loading histories through Equations (12)–(15) by the maximumdeformation (�max/�p) experienced by the beams.

In energy-based seismic design of a steel moment frame, in order to evaluate the seismic behaviorof the entire structure, it is necessary to obtain the energy dissipation capacity of each structuralelement (steel beam in a weak-beam frame). During the design procedure, it is possible to calculatethe maximum deformations of the beams via response analysis. Thereupon, the energy dissipationof the beams can be evaluated through the equations suggested in this paper. Consequently, theenergy dissipation capacity, i.e. the seismic performance of the entire steel moment frame can beevaluated. The results of this study are useful in the design work used to estimate the seismicperformance of steel moment frames.

ACKNOWLEDGEMENTS

Grateful acknowledgment is given to Prof. Keiichiro Suita of the Kyoto University for providing hisexperimental data of steel beams under cyclic loading.

REFERENCES

1. Akiyama H. Earthquake-Resistant Limit-State Design for Buildings. University of Tokyo Press: Tokyo, 1985.2. Krawinkler H. Guidelines for cyclic seismic testing of components of steel structures. Report No. ATC-24,

Applied Technology Council, Redwood City, CA, 1992.3. Clark P, Frank K, Krawinkler H, Shaw R. Protocol for fabrication, inspection, testing, and documentation of

beam-column connection tests and other experimental specimens. SAC Steel Project Background Document,Report No. SAC/BD-97/02, 1997.

4. Federal Emergency Management Agency. Interim testing protocols for determining seismic performancecharacteristics of structural and nonstructural components. FEMA 461 Draft Document, Applied TechnologyCouncil, Redwood City, CA, 2007.

5. Building Research Institute and the Japan Iron and Steel Federation. Study on Testing Method for StructuralPerformance Evaluation of Steel Structures. Tokyo, 2002.

6. Youssef N, Bonowitz D, Gross JL. A survey of steel moment-resisting frame buildings affected by the 1994Northridge earthquake. National Institute of Standards and Technology. NISTIR 5625, 1995.

7. Krawinkler H. Loading histories for cyclic tests in support of performance assessment of structural components.The 3rd International Conference on Advances in Experimental Structural Engineering, San Francisco, 2009.

8. Okada K, Matsumoto Y, Yamada S. Evaluation of effect of joint efficiency at beam-to-column connection onductility capacity of steel beams (in Japanese). Journal of Structural and Construction Engineering, Transactionsof AIJ 2003; 568:131–138.

9. Matsumoto Y, Yamada S, Akiyama H. Fracture of beam-to-column connections simulated by means of theshaking table test using the inertial loading equipment. Behaviour of Steel Structures in Seismic Areas. Balkema:Rotterdam, 2000; 215–222.

10. Kishiki S, Uehara D, Yamada S, Suzuki K, Saeki E, Wada A. New ductile steel frames limiting damage toconnection elements at the bottom flange of beam-ends: part 3 experimental evaluation of composite effects anddamage to concrete slab (in Japanese). Journal of Structural and Construction Engineering, Transactions of AIJ2005; 595:123–130.

11. Suita K. Flexural strength demand for beam-to-column connections in earthquake resistant design of steelmoment frames (in Japanese). Journal of Structural and Construction Engineering, Transactions of AIJ 2003;567:165–171.

12. Kobayashi A. Evaluation of ultimate energy dissipation capacity of steel under cyclic loading (in Japanese).Master Thesis, submitted to Tokyo Institute of Technology; 2005.


1570 Y. JIAO ET AL.

13. Nakagomi T, Fujita T, Minami K, Lee K, Murai M. Study on beam-end detail for the method of non-scallop onbeam-to-column welded joints (in Japanese). Journal of Structural and Construction Engineering, Transactionsof AIJ 1997; 498:145–151.

14. Architectural Institute of Japan. Japanese architectural standard specification JASS 6, steel work (in Japanese).Tokyo: AIJ; 1996.

15. Japanese Standards Association. Japanese Industrial Standard (JIS) Z 2201, Test pieces for tensile test for metallicmaterials, Tokyo, 1998.

16. Kato B, Akiyama H, Uchida N. Ultimate strength of structural steel members (I) (in Japanese). Transactions ofthe AIJ 1966; 119:22–30.

17. Yamada M, Sakae K, Tadokoro T, Shirakawa K. Elasto-plastic bending deformation of wide flange beam–columnsunder axial compression, Part I: Bending moment-curvature and bending moment–deflection relations under staticloading (in Japanese). Transactions of the AIJ 1966; 127:8–14.

18. Yamada S, Akiyama H. Deteriorating behavior of steel members in post-buckling range. Structural Stability andDesign. Balkema: Rotterdam, 1995; 169–174.

19. Architectural Institute of Japan. Recommendation for design of connections in steel structures (in Japanese). AIJ:Tokyo, 2006.

20. Suita K, Tanaka T. Flexural strength of beam web to square tube column joints (in Japanese). Steel ConstructionEngineering 2000; 7(26):51–58.

21. Endo T, Matsuishi M, Mitsunaga K, Kobayashi K, Takahashi K. Rainflow method, the proposal and the applications(in Japanese). Bulletin of the Kyushu Institute of Technology Science and Technology 1974; 28:33–62.


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