EFFECT OF PHASE CHARACTERISTIC OF SEISMIC WAVE ON THE RESPONSE OF STRUCTURES

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1 3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 004 Paper No. 960 EFFECT OF PHASE CHARACTERISTIC OF SEISMIC WAVE ON THE RESPONSE OF STRUCTURES Haibo WANG SUMMARY In seismic resistance design, the design seismic motion is usually defined by elastic response spectra to reflect the characteristic in frequency. Whilst, either the phase of a reference seismic record or random phase is used in generation of artificial wave. The influence of phase on structural response can not be properly defined, as it is often omitted for linear elastic responses. But for the seismic design taking structural damage into account, the phase properties of wave are not negligible. In the study, a determinate method with only two parameters for phase properties of wave has been proposed. This makes it possible to generate a seismic wave, which fits both expected elastic response spectra and expected Demanded Strength Spectra. Similar to the considerations for design elastic response spectra, the level of Demanded Strength Spectra for design can also be determined for the observed records. INTRODUCTION In most seismic resistance design, the design seismic motions are usually defined by elastic response spectra, proposed first by Housner [], which can define the characteristics of the motion for single degree of freedom system of different frequency. Whilst, to get a time history satisfying elastic response spectra, either the phase of a reference seismic record or random phase is used in wave generation. The effect of phase properties cannot be properly reflected, as it is often negligible if only linear elastic responses of structures is considered. However, recent seismic observations [,3] show that seismic motion near the fault can reach or even exceed 000gal in elastic response spectra, and it is uneconomic to design the structure without any damage when attacked by such a strong ground motion, because that the event has very low possibility of occurrence. In recently design criteria [4], the structure will be required to have an elastic performance under middle level ground motion but inelastic performance under very strong level of ground motion, namely, damages within controllable extent at certain non-critical position will be acceptable. Inelastic behavior of structure with damage differs from elastic one very much, for instance, the instantaneous structural frequencies vary with input seismic level. For that reason, phase characteristic of design seismic motion is indispensable for analyses in which the damage of structure is taken in consideration, although no clear definition in present standards. Professor, China Institute of Water Resource and Hydraulic Power Research, CHINA, wanghb@iwhr.com

2 In this study, Demanded Strength Spectra of seismic motion has been used to measure the influence of phase properties on nonlinear structure response. Based upon the relation of phases between the input motion, system transfer function and response of SDF system, two parameters have been introduced to determine the phase properties of artificial wave. One of them changes the motion type from vibration to impulse, subsequently changes the level of Demanded Strength Spectra in whole range of frequency. Another alters the level of Demanded Strength Spectra in low frequency. Similar to the determination of elastic response spectra, the phase properties of many near-field seismic records are used to calibrate two parameters for a design seismic motion. PARAMETERS FOR THE PHASE There are some research works [5,6,7] in earthquake engineering paying attention to the phase characteristics of seismic motion. The physical spectra of real seismic records are usually non stationary, which depends on the mechanism of seismic source, wave path and local geological conditions, etc. and can be properly determined if only these conditions are clear for a scenario earthquake. Commonly used design seismic motion is assumed as a stationary process with an empirical intensity envelope in time history. Ohsaki [5] has revealed that the intensity envelope function of seismic wave is similar to the phase derivative probabilistic distribution. This means that for the same Fourier amplitude spectra, non stationarity of the wave can be determined by the phase properties. The phase function of structural response depends on that of transfer function of structure ψ (ω ) and that of input wave θ (ω ), that is φ = ψ + θ () where ω is the circle frequency. To make the energy of different frequency content not cancel with each other, the difference of phase function of response φ (ω) should be keep within +90º to -90º. For an extrem case θ 0, the input wave becomes an impulse and φ (ω) will change from 0º to 80º. Since the ψ (ω ) depends on the structural frequency, it is impossible to keep the φ (ω) within +90º to -90º for all structures with different natural frequency. Here the change of φ (ω) close to resonant frequency is examined because this frequency content is most dominative for structural response. To reach a larger structural response, the change rate of phase function of input motion should be identical to that of transfer function near the resonant frequency of structure. Considering a single degree of fredom system excited by a support induced motion, the phase function of the system can be expressed as ψ ( λ) = tan 3 ξλ λ + (ξλ) () where λ = ω /ω0, ω 0 and ξ are the eigenfrequency of structure and damping ratio. The change rate of the phase function can be deduced from its derivative, dψ ( λ) dλ 3ξ + 4ξ = ξ ξ λ= (3)

3 That is the change rate of transfer function ψ (ω ) is inversely proportional to the eigenfrequency of structure and damping ratio. dψ dψ dλ dψ = = (4) dω dλ dω ω dλ ω ξ o o To maintain the variation of phase φ(ω) of structural response near resonant frequency within a given range, it is better to make its derivative as small as possible. Then the derivative of the phase of input wave can be decided from Equation and Equation 4, dθ dω = ξω (5) As the phase derivative of the input waves is set to equal to that of transfer function of structure near resonant frequency, the phase of the input waves will change very rapidly at low frequency, the phase of response φ (ω ) which changes from +90º to -90º, is the portion close to resonant point for every structure. Consequently, the phase of input wave is estimated with following equation. a θ = ln( ω + b) + θ 0 ξ (6) in which, a and b are parameters to be determined, they can be so selected that corresponding Demanded Strength Spectra of the generated seismic wave fit the expected ones. In the paper, single degree of freedom system with bi-linear degrading stiffness model has been used to calculate the Demanded Strength Spectra of seismic wave as an example. The initial frequency is determined by system mass and initial stiffness, the initial damping ratio is 5%. The yielding strength coefficient is expressed as the ratio of yielding strength to the weight of the structure. And plastic ratio is the total displacement to yielding displacement. A direct step by step integration based on Newmark s algorithm has been used to calculate the structural nonlinear response. RESULTS AND DISCUSSION Effects of parameter a and b To identify the effects of phase function on nonlinear response of structure, many waves have been generated with identical Fourier amplitude spectra but different phase function through changing the parameter a and b in Equation 6 and the corresponding Demanded Strength Spectra of each wave have been computed. By examining Equation 6 and Figure, it can be found that, as a becomes smaller, the value of phase function varies more slowly, and the energy of seismic wave concentrates within a shorter duration and the peak acceleration of the wave will increase. This implies that the wave transforms from vibration type to impulse type, therefore Demanded Strength Spectra corresponding to the wave become bigger as a whole. This difference affects the elastic response spectra at high frequency too, see the lines in Figure for plastic ratio µ =.

4 From Figure it can be seen that if larger value of parameter b is taken, the phase derivative at lower frequency will become smaller, in consequence, the yielding strength coefficient in Demanded Strength Spectra at lower frequency increases in general. These features of two parameters provide a suitable way to make the generated wave fit an expected Demanded Strength Spectra..0 µ = µ =.0 a= b=0 a= b=0 a=.0 b=0 µ plastic ratio of response µ=8.0 µ = period of structure Figure influence of parameter a on Demanded Strength Spectra.0 µ=8.0 µ=4.0 µ= µ=.0 a=.0, b=0.0 a=.0, b= a=.0, b=5.0 µ plastic ratio of response 0. Period of structure Figure Influence of parameter b on Demanded Strength Spectra

5 Example of generating wave for design Similar to the elastic response spectra for seismic design motion, Demanded Strength Spectra is an important measurement, especially for nonlinear structural response. But up to the present it is very difficult, almost impossible, to define either Fourier amplitude spectra or Fourier phase spectra of seismic motion at a given site for an expected earthquake even supposing the seismic source is clear. Almost all seismic motions for design are based on seismic records of experienced earthquakes as well as similarities in seismic environment between different places. Therefore, it is rationally to use the phase properties of seismic records in past earthquakes to calibrate the phase function. Totally near source seismic records from recent earthquakes, as shown in Table, have been collected. Their phase properties are used to create waves and Demanded Strength Spectra of each wave are computed, as illustrated in Figure 3 and 4. The target Demanded Strength Spectra of design seismic motion here are assumed to be the smallest cover of those generated with the phases of observed records. The Demanded Strength Spectra of achieved seismic motion are displayed in the same figures in red lines, by adjusting two parameters in Equation 6, satisfying overall fit has been achieved. Table Near-source seismic records from recent earthquakes No Earthquake Name of seismic record Max. Acc. (gal) Latitude Longitude Ground Property of soil NS EW level at the position of seismometer Port Island GL-83 Vs=450 (m/s) Takasago Power Station GL-00 Vs=460 (m/s) 3 Hyogoken- SGK,Kansai Elec. Power Co GL-97.0 Vs=455 (m/s) 4 Nanbu Roko (Kobe University) GL-9.5 m (40m/s) layer over Vs=590 (m/s) 5 Inagawa GL-30 Vs=780m/s 6 Great Bridge of East Kobe GL-33 Layer of N=8 above GL-45 7 Coyote Lake San Ysidro GL0.0 Rock 8 Santa Cruz UCSC GL0.0 Limestone Loma Prieta 9 Gilroy#-Gavilan Coll GL0.0 Franciscan Sandstone 0 Landers Joshua Tree fire station GL0.0 Shallow alluvium over granite bedrock Northridge Pacoima Kagel Canyon GL0.0 Rock CONCLUSIONS The influence of phase properties of seismic wave on structural response has been examined. Upon the analyses of the relation between phases of input motion, transfer function of single degree of freedom system and its response, a phase function of two parameters has been introduced. From the example, it is can be found that design seismic wave can be so generated that both elastic response spectra and Demanded Strength Spectra will be well fitted with proposed method in this preliminary study. Although the phase of wave is given in a determinate way in order to fit expected Demanded Strength Spectra with only two parameters, the phase can also defined in a probabilistic way so that non stationarity of seismic waves can be taken into account.

6 .0 plastic ratio 4 by proposed method 0. period of structure Figure 3 Comparison of Demanded Strength Spectra between proposed method and reference seismic records plastic ratio.0 plastic ratio 8 by proposed method 0. period of structure Figure 4 Comparison of Demanded Strength Spectra between proposed method and reference seismic records

7 REFERENCES. Housner G W, et al., "Spectrum analysis of strong-motion earthquakes", Bull. Seismic. Soc. Am Wang, H and Nishimura, A "Determination of design seismic motion by considering inland and interplate earthquakes", Quarterly Report of RTRI, 999,Vol.40, No.3 3. Wang, H and Nishimura, A "On seismic motion near active faults based on seismic records", Proceedings of the th World Conference on Earthquake Engineering, Auckland, New Zealand, Paper No Railway Technical Research Institute, "Standards for Seismic Design of Railway Structures"(in Japanese), 999, Japan 5. Ohsaki, Y. "On the significance of phase content in earthquake ground motions", Earthquake Engineering and Structural Dynamics, 979, Vol.7, Katukura, H. et al., "A study on the Fourier analysis of non-stationary seismic waves" Proceeding of 8 th World Conference on Earthquake Engineering, San Francisco, Vol., Tiliouine B, Hammoutene M And Bard P Y, " Phase angle properties of earthquake strong motions: a critical look" Proceedings of the th World Conference on Earthquake Engineering, Auckland, New Zealand, Paper No