a study on hysteretic plastic energy input into single and multi · pdf file ·...

A study on hysteretic plastic energy input into single and multi degree of freedom systems subjected to earthquakes

K. Sawada1, A. Matsuo1 & K. Ujiie2 1Department of Social and Environmental Engineering, Hiroshima University, Japan 2Omoto Corporations, Japan

Abstract

This paper relates the hysteretic plastic energy inputs of single degree of freedom (SDOF) and multi degree of freedom (MDOF) systems. First, the simple empirical rule between the hysteretic plastic energy input ratio EH/EI and the ratio of the yield strength to the response acceleration spectrum are shown by a large data set of numerical results. Based on this rule, a prediction method of seismic input energy EI and hysteretic plastic energy EH using a design acceleration spectrum is presented. Finally, the time domain analyses of SDOF and MDOF systems under generated earthquake motions fitted to the design spectrum show the validity of the presented method. Keywords: earthquake motion, hysteretic plastic energy, SDOF system, MDOF system.

1 Introduction

Many modern seismic design building codes allow for the plastic deformation in structural members and avoid the catastrophic collapse in severe earthquakes. Seismic design approaches based on the constraint of the maximum plastic deformation of structure have been presented in many previous studies [1, 2]. On the other hand, energy based design concepts have also been proposed. Housner [3] evaluated the total energy input that contributes to a building's responses in elastic and elastic-plastic systems. Akiyama [4] developed Housner's method to devise an earthquake-resistant design method that could be applied in a uniform manner from one-story frames to high-rise buildings. In the

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 269

energy based design methodology, the total energy and the hysteretic plastic energy for the system must be quantified. Then the hysteretic to energy input ratio, EH/EI: that is, the ratio of the hysteretic plastic energy input, EH, to the total energy input, EI must be decided. The approximations of the ratio EH/EI as a function of the damping ratio, a cumulative plastic deformation ratio, η and a ductility factor, µ, have already been presented in previous researches. This paper first deduces the simple empirical rule between the ratio EH/EI, and the ratio of the yield strength to the response acceleration spectrum instead of η and µ from numerical results for single degree of freedom systems subjected to recorded earthquakes. Then, a prediction of seismic input energy, EI, and hysteretic plastic energy, EH, from the design acceleration spectrum is presented for the single and multi degree of freedom systems, based on the above rule. Finally, time domain analyses for earthquake motions fitted to the design spectrum show the validity of the presented method.

2 Energy input concept for SDOF system

Figure 1 shows a viscous damped SDOF system subjected to earthquakes. The equation of motion for the system is expressed as the following equation:

0)( xmxQxcxm −=++ (1) In eqn.(1) m, x, x0, c and Q(x) denote mass, displacement of the mass relative to the ground, the horizontal ground motion, the damping coefficient and the restoring force respectively. Multiplied by x on both sides, and integrated over the entire duration of an earthquake TD, eqn.(1) is reduced to the following equations [4]:

IHEK EEEEE =+++ ξ (2)

0

TD

KE mxxdt= ∫ (3)

2

0

TDE cx dtξ = ∫ (4)

KexQE END

E 2)( 2

= (5)

2

0

( )( )2

TD ENDH

Q xE Q x xdtKe

= ⋅ −∫ (6)

00

TD

IE mx xdt= − ⋅∫ (7)

In eqns.(1)-(7), EK is the kinetic energy, Eξ is the energy input absorbed by damping, EE is the elastic strain energy, EH is the hysteretic plastic energy absorption, and EI is the total earthquake energy input.

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

270 Earthquake Resistant Engineering Structures V

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 1 2 3 4 5

El C entro

Taft

Kobe

Hachinohe

Figure 1: SDOF systems. Figure 2: Normal bilinear hysteretic

model.

Table 1: Recorded earthquakes.

Max. acceleration (cm/s2) Duration time(s) El Centro NS 341.7 53.73 Taft EW 175.9 54.38 Kobe NS 818.0 29.70 Hachinohe EW 182.9 35.99

Figure 3: Acceleration spectrum.

3 The hysteretic to input energy ratio, EH/EI, for SDOF system subjected to recorded earthquakes

This section relates the ratio EH/EI for inelastic systems with a damping proportional to instantaneous stiffness. The recorded earthquakes used in this

D

Ke

Kp

O

A B

E Qy

ｘ δy

C

Q(x)

ｍ

ｘ0

ｘ

Natural Period T0 (s)

2% Damping Acceleration Spectrum (cm/s2)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 271

study are shown in Table 1. The acceleration response spectrums for these waves are shown in fig.3. The hysteretic to energy input ratio EH/EI of a large number of SDOF systems for the combinations of ξ=0.02,0.05,0.10, Qy=0.1Mg, 0.3Mg, 0.5Mg, and the second stiffness Kp=0.01Ke,0.10Ke,0.30Ke subjected to 4 recorded earthquakes are plotted in fig.4 in association with the ratio of the shear yield strength to the elastic acceleration response spectrum Qy/MSA(T,ξ). These results are calculated by the average acceleration method (0.002 seconds integral time interval). By changing the elastic stiffness, the periods of the systems are created from 0.1 seconds to 5.0 seconds in 0.01 seconds intervals. From this figure, the following features are confirmed. (1) The ratio EH/EI decreases linearly with the increase of Qy/MSA(T,ξ). (2) EH/EI approaches 0 as Qy/MSA(T,ξ) approaches 1, and EH/EI approaches 1 as Qy/MSA(T,ξ) approaches 0. The following equation represents the above features:

),(1/

ξTMSAQyEE IH −= (8)

This equation shows the rough average of numerical results in fig. 4.

Figure 4: EH/EI vs. Qy/MSA(T0,ξ). The following equation is presented as the approximate envelope of analysis of results in fig. 4:

EH/EI

Qy/MSA(T0,ξ)

Eqn(8)

Eqn(9)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

272 Earthquake Resistant Engineering Structures V

ββ

ξ

/1

),(1/

−=

TMSAQyEE IH (9)

where β=1.5.

4 Simple prediction of hysteretic plastic energy input from the design acceleration spectrum

4.1 Design acceleration spectrum

The design acceleration spectrum is defined as follows:

≤⋅⋅≤≤⋅⋅

≤≤+⋅⋅=

)5.0(/)(25.1)5.025.0()(5.2

)25.00()61()(),(

TTATA

TTATS A

ξαξα

ξαξ (10)

πξξ 2/),(),( TTSATSV ⋅= (11) 2)2/(),(),( πξξ TTSATSD ⋅= (12)

In eqns.(10)-(12), SA, SV, SD, T and ξ denote the acceleration response spectrum, the velocity response spectrum, the displacement response spectrum, the natural period, and the damping ratio, respectively, and A is 1005(cm/s2). α(ξ) is the reduction factor defined by the following equations [10].

)02.0(/)()( RR ξξα = (13)

{ }78.1/4ln(424.0/4

1)(/4

++−

=−

TTDTTD

eRTTD

πξπξ

ξπξ

(14)

In eqn.(14) TD denotes the duration time of an earthquake motion.

4.2 Artificial waves fitted to the design spectrum

In this study, the five artificial waves fitted to the above-mentioned design spectrum are generated. These waves are used to show the validity of the prediction method described later. The duration of each artificial ground motion is 40.96(s), and uniform random numbers were adopted for the characteristics of phase angles. The following Jennings-type envelope function is employed.

≤≤

≤≤≤≤

=−− )(

)(1)0()/(

)()(

2

TDtTcE

TctTbTbtTbt

tETcta

(15)

where Tb=3.8(s), Tc=19.4(s), a=0.11, TD=40.96(s).

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 273

The velocity design spectrum and the velocity response spectrums of 5 waves are shown in fig.5. Fig.6 shows the design spectrum, eqn.(11), for ξ=0.0, ξ=0.02, and ξ=0.10, and analytical results of the average velocity response for 5 artificial waves. It is confirmed that the design spectrum more or less corresponds to the analytical results.

Figure 5: Velocity response spectrum for artificial waves.

Figure 6: Design spectrum and response spectrum for 5 artificial waves.

0

50

100

150

200

250

300

0 1 2 3 4 5

Velocity Response Spectrum (cm/s)

Natural Period T0 (s)

0

100

200

300

400

500

600

0.0 1.0 2.0 3.0 4.0 5.0

SV(T0,ξ) (cm/s)

Average Response for 5 Artificial Waves

Design Spectrum ξ=0.0

ξ=0.02

ξ=0.10

T0 (s)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

274 Earthquake Resistant Engineering Structures V

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

4.3 The hysteretic to input energy ratio, EH/EI, for an SDOF system subjected to artificial earthquake motions fitted to the design spectrum

The hysteretic to energy input ratio EH/EI of a large number of SDOF systems for ξ=0.02 and 0.05, Qy=0.1Mg, 0.3Mg and 0.5Mg, and the second stiffness Kp=0.01Ke subjected to 5 artificial earthquake motions fitted to the design spectrum are plotted in fig.7 in association with the ratio of the shear yield strength to the elastic acceleration response spectrum Qy/MSA(T,ξ). It is confirmed that eqn.(8) represents the rough average of numerical results and eqn.(9) represents the approximate envelope of numerical results.

4.4 The hysteretic plastic energy input prediction for an SDOF system from design acceleration spectrum

Fig.8 shows the average velocity response spectrums )0.0,( 0TSV and

)02.0,( 0TSV and the average seismic total energy input )02.0,( 0TV eE =

MTE eI /)02.0,(2 0 for an elastic SDOF system under 5 artificial waves and

the design velocity spectrum )0.0,( 0TSV and )02.0,( 0TSV . According to this

figure where )02.0,( 0TV eE is located between )0.0,( 0TSV and )02.0,( 0TSV ,

the following design spectrum for the seismic total energy input, )02.0,( 0TV e

E can be presented:

{ })02.0,()0.0,()02.0,()02.0,( 0000 TSTSTSTV VVVe

E −+= γ (16)

Figure 7: EH/EI vs. Qy/MSA(T0,ξ).

EH/EI

Qy/MSA(T0,ξ)

Eqn(8)

Eqn(9)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 275

0

100

200

300

400

500

600

0.0 1.0 2.0 3.0 4.0 5.0

In eqn.(16), γ is set at 0.7 so that )02.0,5.0(eEV corresponds to numerical

results. Accordingly, the following equation is obtained:

{ })0.0,(7.0)02.0,(3.02

)02.0,( 000

0 TSATSAT

TV eE +=

π (17)

Eqn.(17) is also shown in fig.8. It is observed from this figure that eqn.(17) more or less corresponds to numerical results. The inelastic seismic total energy input spectrum )02.0,( 0TV ep

E can be generated by considering that the instantaneous vibration period is elongated as the inelastic deformation develops. Akiyama presented the following maximum instantaneous vibration period, elongated as the inelastic deformation [4]:

081 TT m

m ⋅

∆+=

η (18)

where Tm is the maximum instantaneous vibration period, ∆ηm is the maximum plastic deformation in a half cycle. In this study, ∆ηm is estimated from the following equation:

{ }1),(/2 0 −⋅=∆ ξη TMSAQym (19) Accordingly, the following equation is obtained:

00

41),(/

1 TTMSAQy

Tm ⋅

−+=

ξ (20)

The inelastic seismic total energy input spectrum, )02.0,( 0TV epE , can be

described by the following equation:

Figure 8: Response spectrum and energy input spectrum.

SV,VEe (cm/s)

T0(s)

Average SV response for 5 artificial wavesAverage VE

e responsefor 5 artificial waves

○

Design velocity spectrum, eqn.(11)

Design VEe spectrum, eqn.(17)

×

SV,(T0,0.0)

SV,(T0,0.02)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

276 Earthquake Resistant Engineering Structures V

0

50

100

150

200

250

300

350

0 1 2 3 4 5

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5

+

++

=

+=

)02.0,2

(7.0)02.0,2

(3.02

)02.0,2

()02.0,(

000

00

TTSA

TTSA

T

TTVTV

mm

meE

epE

π

(21)

The following equation is obtained from eqn.(8).

{ }200

0 )02.0,(21

)02.0,(1)02.0,( TVM

TMSAQyTE ep

EH ⋅

−= (22)

The following equation is obtained from eqn.(21) and eqn.(22).

Figure 9(A): VEep and VEH (Qy=0.1Mg).

VEep,VEH (cm/s)

Qy=0.1m・ g ξ=0.02, Kp=0.01Ke

VEep,Eq.(21) VEH,Eq. (23)

○VEep (Numerical analysis)

●VEH (Numerical analysis)

＋ ＋

VEep,VEH (cm/s)

T0(s)

Qy=0.3m・ g ξ=0.02, Kp=0.01Ke

T0(s)

Figure 9(B): VEep and VEH (Qy=0.3Mg).

VEep,Eq.(21) VEH,Eq. (23)

○VEep (Numerical analysis)

●VEH (Numerical analysis)

＋ ＋

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 277

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

+

++

⋅−=

=

)02.0,2

(7.0)02.0,2

(3.02)02.0,(

1

)02.0,(2)02.0,(

000

0

00

TTSA

TTSA

TTMSAQy

MTE

TV

mm

HEH

π

(23) Figure 9 shows the average response of VE

ep(T0,0.02), VEH(T0,0.02) for 5 artificial waves and eqns.(21) and (23). It is observed from this figure that eqn.(21) and eqn.(23) correspond to numerical results for artificial waves.

4.5 EH/EI for MDOF system subjected to earthquake motions fitted to design spectrum

In this section the ratio EH/EI is derived for the MDOF system shown in fig.10. The average response of the ratio EH/EI of 32 MDOF systems which have different number of stories, stiffness and shear yield strengths subjected to 5 artificial earthquake motions fitted to the design spectrum are plotted in fig.11 in

association with the weighted average of Qyi/QSRSSi, ∑=

⋅n

i iSRSS

ii Q

Qy1

θ .

iSRSSQ and iθ are shown in the following equations:

ｍi

ｘ0

ｘi

ki

ｍn

kn EH/EI

The weighted average of Qyi/QSRSSi

Figure 10: MDOF system. Figure 11: EH/EI for MDOF system.

Eqn.(26)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

278 Earthquake Resistant Engineering Structures V

0

50

100

150

200

250

300

0 0.5 1 1.5 2

∑ ∑= =

=n

j

n

ikjjkjkjiSRSS TSAmuQ

1

2

),( ξλ (24)

where λj is the j-th participation factor, ujk is the j-th eigen mode, and mk is the mass of story k.

∑=

= n

iiiSRSS

iiSRSSi

kQ

kQ

1

2

2

/

/θ (25)

It is confirmed from fig.11 that eqn.(26) follows numerical results well:

∑=

⋅−=n

i iSRSS

iiIH Q

QyEE

11/ θ (26)

4.6 The hysteretic plastic energy input prediction for an MDOF system from design acceleration spectrum

The seismic total energy input for an MDOF system can be predicted from the following equation.

{ }∑=

⋅=n

jj

epEjeI TVmE

1

2)02.0,(

21

(27)

where mej is the j-th effective mass, and Tj is the j-th natural period. The following equations are obtained from eqn.(26) and eqn.(27).

{ }∑∑==

⋅

⋅−=

n

jj

epEje

n

i iSRSS

iiH TVm

QQy

E1

2

1)02.0,(

211 θ (28)

{ }∑∑==

⋅−=

n

jj

epE

jen

i iSRSS

iiEH TV

M

m

QQy

V1

2

1)02.0,(1 θ (29)

SV, VEH (cm/s)

T0 (s)

Figure 12: VEH for MDOF systems.

－Design velocity spectrum�VEH (analysis) ■Eqn.(29)

SV(T0,0.02)

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 279

Fig.12 shows eqn.(29) and the average response of the hysteretic plastic energy input EH under 5 artificial earthquake motions for 12 MDOF systems that have different number of stories, stiffness and shear yield strengths. It is observed from this figure that eqn.(29) corresponds to numerical results well.

5 Conclusions

In this paper, a simple empirical rule between the hysteretic plastic energy input ratio EH/EI and the ratio of the yield strength to the response acceleration spectrum has been shown by a large data set of numerical results for both SDOF and MDOF systems. Based on this rule, a prediction method of seismic input energy EI and hysteretic plastic energy EH has been presented. The time domain analyses under generated earthquake motions fitted to the design spectrum have shown the validity of this new method.

References

[1] A.Shibata and M.A.Sozen, Substitute-Structure Method for Seismic Design in R/C, J. of Struct. Div. ASCE, Vol.102, ST1, 1-18, 1976.1

[2] T.Nakamura, M.Tsuji and I.Takewaki：Design of Steel Frames for Specified Seismic Member Ductility via Inverse Eigenmode Formulation, Comp. & Struct. Vol.47, No.6, 1017-1030, 1993

[3] G.W. Housner, Behavior of Structures During Earthquake, J. of Eng. Mech. Div., ASCE, Vol.85, EM4, 109-129, 1959.4

[4] H.Akiyama：Earthquake-Resistant Limit-State Design for Buildings, University of Tokyo Press, 1985

[5] H.Kuwamura, T.V.Galambos, Earthquake Load for Structural Reliability, J. of Structural Engineering, ASCE, Vol.115, ST6, 1989.6

[6] P.Fajfar, T.Vidic, Consistent Inelastic Design Spectra : Hysteretic and Input Energy, Earthquake Engineering and Structural Dynamics, Vol.23, 523-537, 1994

[7] L.D.Decanini, F.Mollaioli, An Energy-based Methodology for the Assessment of Seismic Demand, Soil Dynamics and Earthquake Engineering, 21, 113-137, 2001

[8] M.F. Cruz A, O.A. Lo´pez, Plastic Energy Dissipated during an Earthquake as a Function of Structural Properties and Ground Motion Characteristics, Engineering Structures 22, 784–792, 2000

[9] Architectural Institute of Japan, Recommendations for Loads on Buildings, 1993 (in Japanese)

[10] E.Rosenblueth, J.I.Bastamante, Distribution of Structural Response to Earthquakes, J. of Engineering Mechanics Div., ASCE, Vol.88, EM3, 1962.6

[11] H.Kuwamura, Y.Kirino and H.Akiyama, Prediction of Earthquake Energy Input from Smoothed Fourier Amplitude Spectrum, Earthquake Engineering and Structural Dynamics, Vol.23, 1125-1137,1994

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

280 Earthquake Resistant Engineering Structures V