Simulations of non-stationary frequency content and its importance to seismic assessment of...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 31:993–1005 (DOI: 10.1002/eqe.134)

Simulations of non-stationary frequency content and itsimportance to seismic assessment of structures

Junjie Wang1;∗;†, Lichu Fan1, Shie Qian2 and Jing Zhou3

1State Key Laboratory for Disaster Reduction in Civil Engineering; Tongji University; Shanghai 200092; China2National Instruments; 11500 N Nopac Expwy Building B; Austin; Texas 78759-3504; U.S.A.

3Department of Civil Engineering; Dalian University; Dalian 116023; China

SUMMARY

To verify the importance of the non-stationary frequency characteristic of seismic ground motion, ajoint time–frequency analysis technique of time signals, called chirplet-based signal approximation, isdeveloped to extract the non-stationary frequency information from the recorded data. The chirplet-basedsignal approximation is clear in concept, similar to Fourier Transform in mathematical expressions butwith di?erent base functions. Case studies show that the chirplet-based signal approximation can repre-sent the joint time–frequency variation of seismic ground motion quite well. Both the random models ofuniform modulating process and evolutionary process are employed to generate artiAcial seismic waves.The joint time–frequency modulating function in the random model of evolutionary process is deter-mined by chirplet-based signal approximation. Finally, non-linear response analysis of a SODF systemand a frame structure is performed based on the generated artiAcial seismic waves. The results showthat the non-stationary frequency characteristic of seismic ground motion can signiAcantly change thenon-linear response characteristics of structures, particularly when a structure goes into collapse phaseunder seismic action. It is concluded that non-stationary frequency characteristic of seismic ground mo-tion should be considered for the assessment of seismic capacity of structures. Copyright ? 2002 JohnWiley & Sons, Ltd.

KEY WORDS: non-stationary frequency; chirplet-based signal approximation; non-linear response;seismic assessment

1. INTRODUCTION

The time history analysis is commonly used for seismic designs, veriAcation of seismic ca-pacity and seismic assessment of structures, because of the signiAcant progress in the Aeld of

∗ Correspondence to: Junjie Wang, State Key Laboratory for Disaster Reduction in Civil Engineering, TongjiUniversity, 1239 Sipin Road, Shanghai 200092, China.

† E-mail: [email protected]

Contract=grant sponsor: National Science Foundation of China; contract=grant number: 40072088.

Received 4 January 2001Revised 14 June 2001 and 9 August 2001

Copyright ? 2002 John Wiley & Sons, Ltd. Accepted 10 August 2001

994 J. WANG ET AL.

0 5 10 15 20 25 30 35 40 45-1.4

-0.7

0.0

0.7

1.4

Acc

. (m

/s2 )

Time (sec)

Figure 1. A typical record of strong seismic ground motion.

Anite element method. Seismic input for the time history analysis is the time history of seismicground motion, which can be obtained by the following methods: (1) select a recorded historyof seismic ground motion during the past earthquake events according to some requirements(site type, and epicentre distance, etc.); (2) generate artiAcial seismic waves by a theoretical(or empirical) model of seismic ground motion.Owing to the lack of recorded data and the randomness of earthquake ground motion that

may be experienced by the structure in the future, usually it is diGcult to obtain recordeddata which At the requirements (site type, and epicentre distance, etc.) well. Therefore theartiAcial seismic waves are widely used in seismic designs, veriAcation of seismic capacityand seismic assessment of structures.Currently, the most commonly used random model to generate an artiAcial seismic wave is

written asa(t)=f(t) x(t) (1)

where f(t) is a deterministic absolute amplitude envelope function with a maximum valueof 1, and x(t) is a stationary random process with a power spectral density function S(!).Equation (1) is traditionally called a uniformly modulated random process, which describes aclass of random processes with non-stationary intensity but time-invariant (constant) frequencycontent.Figure 1 illustrates a typical earthquake accelerogram (N–S component, comes from

SMART-1 in Luotong, Taiwan, the magnitude is Ms = 7:0, epicentre=79 km). Measuredground motions are non-stationary time series. The non-stationarity is manifested in two as-pects. First, the intensity of the ground acceleration varies with time; after arrival of theArst seismic waves, it builds up rapidly to a maximum value over several seconds and thendecreases more slowly until it vanishes into background noise. Second, the frequency con-tent varies with time, with a tendency to shift to lower frequencies as time increases. Thesenon-stationarities can be attributed partly to the di?erent intensities, frequency content, andarrival times of the P (primary or ‘push’), S (secondary or shear), and surface (Rayleighand Love) waves, and partly due to the Anite rupture time and Anite fault area. Obviously,Equation (1) has included the amplitude non-stationarity but has neglected the temporal changeof the frequency content.Recently, researchers have studied the time-varying frequency characteristics of earthquake

ground motion, including modiAcation of stochastic models [1–4]. The signiAcance of time-varying frequency characteristics to the seismic response of structures has also been reportedin the literature [3; 5–7]. It was commonly believed that the time-varying frequency charac-teristics are important for the inelastic response of structures.

Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:993–1005

SIMULATIONS OF NON-STATIONARY FREQUENCY CONTENT 995

In this paper, a fully non-stationary ground motion model with both intensity and frequencybeing non-stationary is employed to verify the importance of the non-stationary frequencycharacteristic of seismic ground motion to seismic response and seismic capacity assessmentof structures. The content includes: (1) a joint time–frequency analysis technique of time sig-nals, called chirplet-based signal approximation, which is developed to extract the informationof time-varying frequencies from seismic ground motion data samples. Based on that informa-tion, the frequency-dependent modulating function in the stochastic model of the evolutionaryrandom process is computed; (2) the artiAcial waves based on stochastic models of uniformmodulating random process and evolutionary random process, are generated and compared,separately and (3) the importance of time-varying frequency characteristics of earthquakeground motion to the non-linear responses and seismic capacity assessment of structures isinvestigated.

2. FULLY NON-STATIONARY SEISMIC GROUND-MOTION MODEL

As mentioned earlier, the currently used earthquake ground-motion model in Equation (1)failed to characterize the time-varying nature of frequencies in seismic actions. To overcomethis shortcoming, the so-called evolutionary random process model was proposed [8], that is,

a(t)=∫ ∞

−∞A(t; !)ei!t dF(!) (2)

where i=√−1, and A(t; !) is a deterministic frequency–time modulating function. It repre-

sents an absolute amplitude envelope of a seismic process. dF(!) is a zero-mean, mutuallyindependent, orthogonal increment process with

E[dF(!)] = 0

E[dF∗(!1) dF(!2)] = �(!1 −!2)S(!1) d!1 d!2

where E[:] denotes the ensemble average, �(:) is the Dirac delta function, ‘∗’ stands forthe complex conjugate, and S(!) is the power spectral density function of dF(!).It is interesting to note that when A(t; !)≡f(t), Equation (2) reduces to Equation (1). In

other words, Equation (1) is a special case of Equation (2) while Equation (1) is limited tothose processes whose frequency content does not change with time, Equation (2) applies formuch more general cases, such as the time-varying frequency process.In earthquake engineering applications, the frequency–time modulating function A(t; !) only

makes sense when it is real and non-negative, that is,

A(t; !)∈R and A(t; !)¿0

A(t; !) determines only the shape of the intensity envelope (for di?erent frequency) of seismicground motion if it is normalized as

maxfor all t

[A(t; !)]=1 for a given !


996 J. WANG ET AL.

while the amplitude (for di?erent frequencies) of seismic ground motion is controlled byS(!).Based on Equation (2), the following trigonometric method is employed to generate artiAcial

seismic waves [14]

a(t)=n∑k=1A(t; !k)

√2S(!k)O! cos(!kt + �k) (3)

where O! is the frequency increment, !k is the discrete frequency, and �k is the randomphase with uniform distribution between 0 to 2�. Obviously, the key point here is to estimatethe function A(t; !).

3. ADAPTIVE CHIRPLET SIGNAL APPROXIMATION AND ADAPTIVESPECTROGRAM

To determine the function of A(t; !), a joint time–frequency analysis technique of time signals,called chirplet-based signal approximation, is developed in this paper. Unlike those well-knownjoint time–frequency representations, such as the Gabor expansion and wavelets, and time-dependent spectra, such as short-time Fourier transform and Wigner–Ville distribution, theadaptive chirplet signal approximation and adaptive spectrogram are model-based joint time–frequency analysis methods. For a given signal s(t), the adaptive chirplet signal approximationis deAned (Qian and Chen [9]) as

s(t)=K∑k=0pk(t) + sK+1(t) (4)

Note that

s0(t)≡ s(t)The function pk(t) is the regular chirp function with Gaussian envelope, i.e.,

pk(t)=Ake−�k (t−tk )2cos

[!k(t − tk) + �k

2(t − tk)2 + ’k

](5)

Therefore, it is commonly known as the chirplet. The parameters (Ak; �k ; tk ; !k ; �k ; ’k) arechosen such that the mean square error between pk(t) and sk(t) is a minimum. Numeri-cal simulations indicate that by adaptive chirplet approximation, we can use fewer chirpletpk(t) to approximate seismic signals than in the case of either Gabor expansion or waveletrepresentation.Applying the Wigner–Ville distribution, we can further derive the adaptive spectrogram, i.e.,

AS(t; !)≈K∑ 2�

∼ (Ak)2e−�k (t−tk )2−[!−!k−�k (t−tk )]2=�k (6)

Obviously, it is real and non-negative. In particular, owing to higher time–frequency resolution,we chose the adaptive spectrogram to represent the seismic signal energy distribution in thejoint time–frequency domain. Figure 2 illustrates an adaptive spectrogram (which correspondsto Figure 1). Apparently, the frequency content of a seismic signal changes with time.



Figure 2. An example of adaptive spectrogram of seismic records.

4. ARTIFICIAL SEISMIC WAVES

The data used in this paper were collected from the SMART-1 (Strong Motion Array inTaiwan number 1) Array, located in Luotong, Taiwan. The array consists of three concentricrings, with diameters of 200, 1000, and 2000m. On each ring, 12 accelerometers are equallydistributed. In addition, there is one in the centre, making a total of 37 sensors. In thispaper, we selected data samples from 32 of the accelerometers. The data were recordedfrom the 45th seismic event experienced by the SMART array in 1986. The magnitude isML =7:0. The epicentre distance was 79km, at a depth of 7:0km. Figure 3 shows the resulting

Figure 3. Statistical adaptive spectrogram of 32 seismic records.


998 J. WANG ET AL.

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 50 .0

0 .5

1 .0 0 .3 H z

1 .0 H z

4 .0 H z

T ime (se c )

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 50 .0

0 .5

1 .0

0 .5 H z

1 .5 H z

6 .0 H z

T ime (se c )

0 5 10 15 20 25 30 35 40 450.0

0.5

1.0

0.7Hz

2.5Hz

8.0Hz

A(t

,w)

A(t

,w)

A(t

,w)

T ime(sec)

Figure 4. Frequency-dependent modulating function.

frequency-dependent modulating function A(t; !), which is the mean of 32 adaptive spec-trograms of the corresponding seismic signals. For the purpose of this paper, the adaptivespectrograms of the corresponding recorded seismic signals are normalized as the follows:

A(t; !k)=AS(t; !k)/Maxfor all t

(AS(t; !k)); k=1; 2; : : : (7)

In which, k is the discrete frequency point.Figure 4 illustrates time plots of A(t; !) at di?erent frequencies. Obviously, for di?erent !k ,

seismic ground motion amplitudes have rather di?erent patterns. In all three cases, as timeincreases, the low-frequency contents become more and more prominent. This is consistentwith the observation from the recorded data, such as the one in Figure 1. While a uniformmodulating random process based on Equation (1) treats all frequencies equally, the frequency-dependent modulating function A(t; !) emphasizes the e?ect of low-frequency components inlater seismic actions.After obtaining A(t; !), we apply Equation (3) to generate the artiAcial seismic waves.

Figure 5 depicts S(!) and f(t) (in Equation (1)) used in this paper. The examples shownin Figures 6 and 7 use the same A(t; !), S(!), and f(t), but di?erent random phase vector�k . While Figures 6(a) and 7(a) represent the artiAcial seismic waves based on Equation (1);Figures 6(b) and 7(b) are generated by Equation (2).Compared to the time series in Figure 1, it is obvious that the artiAcial waves generated by

Equation (2), such as in Figures 6(b) and 7(b), are much closer to the recorded seismic datathan those based on Equation (1). Basically, there is no frequency change in Figures 6(a)



0 2 4 60.00

0.01

0.02

0.03

Po

wer

Sp

ectr

um

Frequency(Hz)

0 5 10 15 20 25 30 35 40 450.0

0.5

1.0

En

velo

pe

Time (sec)

Figure 5. Power spectrum density S(!) and envelope function f(t).

0 5 10 15 20 25 30 35 40 45-1.4

-0.7

0.0

0.7

1.4

Acc

. (m

/s2 )

Time (sec)

Time (sec)

(a) Uniformly Modulated Model

0 5 10 15 20 25 30 35 40 45-1.4

-0.7

0.0

0.7

1.4

Acc

. (m

/s2 )

(b) Evolutionary Model

Figure 6. The Arst group of artiAcial seismic waves.

0 5 10 15 20 25 30 35 40 45-1.4

-0.7

0.0

0.7

1.4

Acc

. (m

/s2 )

Time(sec)

(a) Uniformly Modulated Model

0 5 10 15 20 25 30 35 40 45-1.4

-0.7

0.0

0.7

1.4

Acc

. (m

/s2 )

Time(sec)

(b) Evolutionary Model

Figure 7. The second group of artiAcial seismic waves.


1000 J. WANG ET AL.

and 7(a), the artiAcial seismic waves based on Equation (1). On the other hand, Equation (2)based artiAcial seismic waves in Figures 6(b) and 7(b) clearly show that the time series startwith high frequencies and are gradually dominated by low frequencies. Since the later seismicaction plays an important role in structure damage, it is rather signiAcant to preserve thoselow-frequency features in the artiAcial input signal.

5. EXAMPLES OF STRUCTURAL RESPONSE

In the preceding sections, we applied the adaptive spectrogram for the fully non-stationaryearthquake ground-motion model. In what follows, we shall investigate the importance ofpreserving the property of time-varying frequency for artiAcial seismic waves.

Example 1 (Response of an oscillator with a single degree of freedom)The example analyzed in this section is an oscillator with a single degree of freedom andperfect elastic–plastic force–displacement, as shown in Figure 8; m, c, and k denote the mass,damping, and sti?ness of the oscillator; Fy is the yield strength (force), m=9:0× 104 kg;c=4:5× 103 Ns=m, and k=1:49× 105 N=m in this study. The response of the oscillator underseismic actions for three values of Fy is shown in Figures 9(a)–(c). The dashed line standsfor the response under artiAcial seismic wave generated from Equation (1) and the solid linestands for that generated from Equation (2).Figure 9 indicates that the solid line and the dashed line get further apart as the yielding

force (Fy) decreases and as the time increases, that is, larger di?erences between the solidlines and dashed lines appear in Figures 9(a)–(c). As we know from the vibration theoryof structures, the response of the oscillator is mainly excited by the frequency contents (ofthe seismic waves) nearly at the free vibration period of the oscillator. The free vibrationperiod of the oscillator increases (the sti?ness of the oscillator decreases) after the yield (i.e.,the internal force response exceeds Fy) and the low-frequency contents of the seismic groundmotion contributes more to the response of the oscillator, that is the moving resonance e?ect.The above is the reason why larger di?erences between the solid lines and dashed lineswere observed from Figures 9(a)–(c). In fact, the structures are usually damaged (some ofthem are seriously damaged) during a strong earthquake, and their free vibration periods (ata given time) become longer with its response history, which coincides qualitatively withthe change (with time) of the frequency spectrum of seismic ground motion. Therefore the

(a) The Oscillator System (b) Force-Displacement Relationship

m

c

k

u(t) a(t) force

displacement

Fy

Fy

Figure 8. The oscillator and mechanical characteristics.



0 5 10 15 20 25 30 35 40 45-0.34

-0.17

0.00

0.17

0.34

Dis

p. (

m)

Time(sec)

F = Ny 30 104.(a)

0 5 10 15 20 25 30 35 40 45-0.34

-0.17

0.00

0.17

0.34

Dis

p. (

m)

Time(sec)

F = N2 . 0 10 4(b)

0 5 10 15 20 25 30 35 40 45-0.34

-0.17

0.00

0.17

0.34

Dis

p. (

m)

Time(sec)

F = Ny

y

1.5 104(c)

×

×

×

Figure 9. The displacement response of the oscillator under seismic action.

9.14 7.31 7.31

[email protected]

4.77

Figure 10. Three-bay and ten-storey frame structure (unit: m).

non-stationary frequency characteristic of seismic ground motion may have a signiAcant e?ecton the non-linear response of the structures and cannot be neglected in the aseismic designof structures.

Example 2 (Three-bay and ten-storey frame)A frame structure, (see Figure 10) analysis is given in Reference [10] and is taken to verifythe e?ects of non-stationary frequency characteristic on the response of structures with many


1002 J. WANG ET AL.

Force

Disp. Moment

Axial Force

My+My-

Pyc

Pyt

(a) Force - Displacement Relationship (b) The shape of Yielding Surface

Figure 11. The features of mechanics of the beam–column element.

Dis

p. E

nvel

ope

Solid : UniformDash : Evolutionary

(a) 0.1g (b) 0.2g

(c)0.3g

number of the floor from the bottom to the top

Figure 12. Envelopes of displacement responses (cm).

Max

. Pla

stic

Rot

atio

ns

(a)0.1g

(b)0.2g

(c)0.3g


Figure 13. Maximum plastic rotations at left ends of beams of the Arst bay (Rad).

degrees of freedom. The structural parameters used in the analysis are given in Reference [10].The features of mechanics of the beam–column element are shown in Figure 11.To verify the e?ects of the non-stationary frequency on the response of the frame structure,

the peak value of the artiAcial wave is adjusted to three levels: (a) 0:1 g; (b) 0:3 g; (c) 0:5 g.The results of structural responses are illustrated in Figures 12–16.It can be observed from Figures 12–16 that the non-stationary frequency of the seismic

ground motion increases signiAcantly the response displacements at the Toors and the plasticrotations at the ends of the beams. The increment may reach more than 50–100 per cent.SigniAcant increments of moments and shear forces at the Arst Toor were observed. Accumu-lated displacement responses were observed as shown in Figure 16(c), with the non-stationary



(a)0.1g (b)0.2g (c)0.3g

Env

elop

e of

Neg

tive

Mom

ent


Figure 14. Envelopes of the negative moments of the fourth columns (kNm).

(b)0.2g(a)0.1g (c)0.3g

Envelo

pe o

f P

osi

tive S

hear


Figure 15. Envelopes of the positive shear force of the fourth columns (kN).

Solid Line: UniformDash Line: Evolutionary

(a)0.1g

(b)0.2g

(c)0.3g Dis

plac

emen

t Res

pons

e H

isto

ries

Time (Sec)

Figure 16. Displacement response histories at the top Toor (cm).

frequency leading to larger accumulated displacements than those given without considerationof the non-stationary frequency characteristic of seismic ground motion.The above-computed results show that the feature of non-stationary frequency of seismic

ground motion is of practical importance to the reasonable prediction of seismic responses


1004 J. WANG ET AL.

and the seismic capacity assessment of structures, particularly for the collapse assessment ofstructures under seismic action. The results presented in this paper strongly suggest a need tomodify current methods for generating artiAcial seismic waves based on Equation (1), whichshould be replaced by a more reasonable random process model, for example, Equation (2).

6. CONCLUSIONS

In this paper, a joint time–frequency technique, called chirplet-based signal approximation,is developed to extract the non-stationary frequency information from the recorded data ofseismic ground motion, and consequently the frequency-correlated modulating function of theevolutionary random process model is determined. Two groups of artiAcial seismic waveswere generated by using the uniformly modulated random process model (Equation (1)) withonly non-stationary intensity included and evolutionary random process model (Equation (2))with both non-stationary intensity and non-stationary frequency included. Finally, the e?ects ofnon-stationary frequency behaviour on the response of non-linear structures were investigated.We have reached the following conclusions:

• The chirplet-based signal approximation is an ideal candidate for extracting the jointtime–frequency information from seismic records. The resulting frequency-dependentmodulating function more accurately reTects the non-stationary frequency characteris-tic of seismic ground motion.

• Using non-stationary frequency information, we can substantially improve the artiAcialseismic waves, in terms of preserving the non-stationary intensity and frequency content,which are presented in recorded accelerograms.

• The non-stationary frequency characteristic of seismic ground motion may have an im-portant e?ect on the non-linear response of structures, including displacement responsesand internal force responses. The results show that to reach a reasonable assessment ofseismic capacity (including deformation capacity and strength) of structures if the ar-tiAcial seismic waves are employed, the non-stationary frequency characteristic shouldbe included in the generation of artiAcial seismic waves along with the non-stationaryintensity of the seismic ground motion.

ACKNOWLEDGEMENT

The authors are grateful for the support of the National Science Foundation of China through GrantNo. 40072088.

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3. Yeh C-H, Wen YK. Modeling of non-stationary ground motion and analysis of inelastic structural response.Structural Safety 1990; 8(1–4):281–298.



4. Papadimitriou C, Bech JL. Non-stationary stochastic characterization of strong-motion accelerograms.Proceedings of the Fourth National Conference on Earthquake Engineering, Vol. I. Earthquake EngineeringResearch Institute, El Cerrito, CA, 1990; 477–486.

5. Conte JP. E?ects of earthquake frequency non-stationarity on inelastic structural response. Proceedings of the10th World Conference on Earthquake Engineering, A.A. Balkema: Rotterdam, The Netherlands, 1992.

6. Beck JL, Papadimitriou C. Moving resonance in non-linear response to fully non-stationary stochastic groundmotion. Journal of Probabilistic Engineering Mechanics 1993; 8(3–4):157–167.

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8. Nigam NC. Introduction to Random Vibrations. MIT Press: Cambridge, Mass, 1983.9. Qian S, Chen D. Joint Time–Frequency Analysis. Prentice-Hall: Englewood Cli?s, NJ, 1996.

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11. Conte JP, Pister KS, Mahin SA. Non-stationary ARMA modeling of seismic motions. Journal of Soil Dynamicsand Earthquake Engineering 1992; 11(7):411–426.

12. Conte JP, Peng BF. Fully non-stationary analytical earthquake ground-motion model. Journal of EngineeringMechanics, ASCE 1997; 123(1):15–24.

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