Time-frequency response spectrum of rotational ground motion and its application∗
Wei Che 1 and Qifeng Luo 2,
1 School of Engineering & Technology, China University of Geosciences, Beijing 100083, China 2 Shanghai Institute of Disaster Prevention and Relief, Tongji University, Shanghai 200092, China
Abstract The rotational seismic motions are estimated from one station records of the 1999 Jiji (Chi-Chi), Taiwan, earthquake based on the theory of elastic plane wave propagation. The time-frequency response spectrum (TFRS) of the rota-tional motions is calculated and its characteristics are analyzed, then the TFRS is applied to analyze the damage mechanism of one twelve-storey frame concrete structure. The results show that one of the ground motion components can not reflect the characteristics of the seismic motions completely; the characteristics of each component, especially rotational motions, need to be studied. The damage line of the structure and TFRS of ground motion are important for seismic design, only the TFRS of input seismic wave is suitable, the structure design is reliable.
The ground motions consist of six components, namely, three translational components and three rota-tional components. Translational components were the focus and rotational components were often neglected in previous studies. But with an in-depth research, the rota-tional components and the structural damage caused by it gradually become the hotspot in earthquake engineer-ing in recent decades. Hart et al (1975) analyzed the ambient and earthquake response records obtained in several southern California high-rise buildings from the San Fernando earthquake. In general, the plane waves incident upon the free surface lead to three translational and three rotational components of ground motions (Trifunac, 1982).
In 2009 the Bull Seism Soc Amer published a spe-cial issue on rotational seismology and engineering ap-plications including 51 papers in three sections. Lee et al (2009) introduced the main sections of this special issue and made comment on each paper briefly and they con- ∗ Received 17 November 2009; accepted in revised form 18 December 2009;
published 10 February 2010. Corresponding author. e-mail: [email protected]
cluded that earthquake monitoring cannot be limited to measuring only the three translational components. Tri-funac (2009) made review on rotations in structural re-sponse, he considered that recording the rotational components of motion will contribute significantly to information on structural response and recommended deployment of instruments to measure rotational com-ponents of motion in free-field and full-scale structures. The tutorial given by Kozák (2009) summarized the historical examples of earthquake rotational effects. Knopoff and Chen (2009) showed that the single-couple equivalent does not violate principles of Newtonian mechanics because the torque imbalance in the single couple is counterbalanced by rotations within the fault zone; the crack therefore radiates torque waves and a rotational deformation field.
According to the results mentioned above, the tor-sional response caused by the rotational ground motions has been an important indicator of earthquake response. Thus rotational ground motions should be considered, especially in seismic design for large-scale complex structures.
Seismic analysis shows that the duration of input time history of ground motion, which reflects the cumu-lative effect caused by strong motion, is an important
Doi: 10.1007/s11589-009-0078-2
72 Earthq Sci (2010)23: 71−77
influence factor for the damage of structure. Because the two-dimensional (2D) acceleration response spectrum cannot account for the characteristic, the time-frequency response spectrum (TFRS) is introduced, which can consider the three characteristics of ground motions (in-tensity, spectrum and duration) and can be used to ana-lyze the damage mechanism of structure (Luo and Zhang, 2002; Luo and Li, 2004, 2005). It is obvious that TFRS is a three-dimensional spectrum.
Huang (2003) inferred the rotational motions from a seven-element accelerometer array on the Li-Yu-Tan dam, located 6 km north of the northern end of the 1999 Jiji (Chi-Chi) earthquake (MW7.6) fault rupture. Because Taiwan is at present the only area that has deployed both translational and rotational seismometers at several sta-tion sites for monitoring regional and local earthquakes (Lee et al, 2009), so we need not estimate rotational component by three seismic translation component re-cords from a single seismographic station. In this paper, we synthesize the rotational motion basing on the trans-lational acceleration records of the Jiji (Chi-Chi) earth-quake from only one station. Then the time-frequency response spectrum (TFRS) of the rotational component
is calculated to analyze its distribution characteristic on time and period axes. Finally the rotational damage mechanism of one twelve-storey frame concrete struc-ture is discussed by TFRS of the rotational motion.
2 Synthesis of rotational ground motion The Jiji (Chi-Chi), Taiwan, MW7.6 earthquake on
20 September 1999 resulted in a 100 km long surface rupture in the north-south direction. This is the largest inland earthquake in the past 100 years in Taiwan, and its mechanism was a thrust fault with a strike of nearly N20°E and a dip of about 20° to 35°. The largest dis-placements concentrated near its northern end of the causative fault (Huang, 2003). The accelerations of the Jiji (Chi-Chi) earthquake were well recorded at free field stations. In this study, three translational (two horizontal and one vertical) acceleration components, which were recorded at station TCU084 (PEER Strong Motion Database, 2009), are used to synthesize the rota-tional ground motions and three translational waves are shown in Figure 1.
Figure 1 The translational waves of the Jiji (Chi-Chi) earthquake recorded at the station TCU084.
The reliable information on rotational ground mo-tions is unavailable, primarily because the rotational motions can not be directly measured. At present, the rotational seismic motions are usually estimated from translational ground motions in terms of the correlation between different components. It is considered that the seismic ground motion is generated by plane harmonic waves arriving at the site. The propagation direction of the seismic waves is assumed to lie in the vertical plane (x, z). The coordinate system is shown in Figure 2. Sup-posing that the seismic waves propagate in a homoge-neous isotropic elastic half space or a layered elastic half space and the near field seismic motion is produced by body waves, the relation between the Fourier spectrum
of rotational motion and that of translational motion is written as (Wang, 1995; Sun and Chen, 1998)
( )( ) i2 ( )gzvc f
ωφ ω ω=&&&& (1)
and
( )( ) i ,( )gy
wc f
ωφ ω ω= −&&&& (2)
where ( )gzφ ω&& and ( )gyφ ω&& are the Fourier spectrum of
the torsional and the rocking acceleration component, respectively; i= 1− ; c( f ) denotes the apparent velocity of the body waves; ( )v ω&& and ( )w ω&& represent the Fou-rier spectrum of out-of-plane and in-plane (vertical di-
Earthq Sci (2010)23: 71−77 73
rection) acceleration component, respectively. Through Fourier inverse transform, the time history of the rota-tional components can be obtained. The method consid-ers the dispersion of seismic waves caused by the propagation in inhomogeneous medium.
Figure 2 The coordinate system.
The apparent velocity c( f ) of the body wave is es-timated by (Sun and Chen, 1998)
2)(lg814.0lg646.1426.5)( fffc −+= (3) and ( ) ( )(1 0.2 ),c f c f ζ= + (4)
where f is frequency in Hz, and ζ is a random number in (−1, 1).
The acceleration time histories of the rotational components, which are synthesized from the three trans-lation components of the Jiji (Chi-Chi) earthquake, are shown in Figure 3. The peak acceleration of the tor-sional component is 0.07 rad/s2. The peak torsional component is larger than the rocking component. How-ever, the peak values of all the components appear at about 40 s on the time axis and the relative large values mainly distribute between 34 s and 50 s, after that the seismic motions decay quickly. It implies that the enor-mous energy of the Jiji (Chi-Chi) earthquake was re-leased in 16 s.
Figure 3 The acceleration time history of the rotational motion. (a) gzφ&& torsional acceleration component;
(b) gyφ&& rocking acceleration component.
3 TFRS of rotational motion The concept of the time-frequency response spec-
trum (TFRS) is the absolute value of response of a series of single degree of freedom systems with same damping and different period over the entire time histories. As a three dimensional spectrum, TFRS is the function of not only period but also time (Luo and Li, 2004, 2005). The flow of TFRS calculation (Figure 4) is: firstly choose a proper damping ratio for the single degree of freedom system; secondly choose the periodic sequence, namely, a series of the single degree of freedom systems with certain natural period; thirdly calculate the seismic re-sponses of the single degree of freedom systems by nu-merical integration method; fourthly make the absolute
Figure 4 The flow chart of time-frequency response spectrum calculation.
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seismic responses form the time-frequency response matrix, which is the function of two variants, i.e., time and period; finally, draw the stereogram and contour diagram of TFRS. The stereogram can reflect the distri-bution characteristics of peak values and the seismic energy in frequency domain and time domain.
The three-dimensional TFRS of the rocking com-ponent and its contour diagram are shown in Figure 5. In time domain, most of the large values are distributed from 35 s to 45 s and the peak value appears at 37 s.
In the period domain, the period of the main por-tion of the TFRS is smaller than 1 s and the peak value appears at about 0.45 s, beyond which the amplitude
decreases with period. It implies that the spectrum of the rocking component distributes in a relative narrow range on the period axis and the low building (two- or three-storey), which has a short natural period, will produce larger response.
The three-dimensional TFRS of the torsional com-ponent and its contour diagram are shown in Figure 6. On the time axis, the peak value appears at 36 s and there are several large values distributed from 37 s to 50 s; after that the amplitude decreases quickly with time. On the period axis, most of the large amplitudes are distributed from 0.5 s to 1.5 s and the peak value appears at 0.9 s.
Figure 5 Time-frequency response spectrum of the rocking component. (a) Stereogram; (b) Contour diagram.
Figure 6 Time-frequency response spectrum of the torsional component. (a) Stereogram; (b) Contour diagram.
Figure 6 can be applied to explain the damage mechanism of the structure. Once the structure is seri-ously damaged by the torsional motion before 37 s in time domain, its natural period will be lengthened. Just then the structure meets the other large amplitude ap-pearing from 0.5 s to 1.5 s on the period axis and from 42 s to 45 s on the time axis, during which the dominant period of the torsional component is consistent with the
changed natural period of the structure, resonance may occur. Even if the natural period of the structure keeps unchanged after 37 s, the structure will also be sub-jected to impact of a series of large torsional accelera-tions, which can also be seen in Figure 7 that gives the cross section of TFRS shown in Figure 6a at 0.9 s on the period axis. From the above analyses we can get that the accumulation damage of the structure cannot be
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Figure 7 The cross section of time-frequency response spectrum shown in Figure 6a at 0.9 s on the period axis.
neglected. Because the torsional and rocking components are
estimated from the translational components recorded at the same station TCU084, there are the same character-istics in time domain, namely, the energy concentrates
in a narrow time range and their peak values appear at about 37 s on the time axis. However, comparison be-tween Figures 5 and 6 indicates the torsional component is richer than the rocking component in period domain, which can also be seen in the two-dimensional response spectrum (Figure 8). As shown in Figure 8, one of the ground motion components can not reflect the charac-teristics of the seismic motions completely. The struc-tures will be subjected to the effect of six components at the same time; hence the characteristics of each com-ponent of ground motions, especially rotational motions, need to be further studied.
Figure 8 Two-dimensional acceleration response spectrum of translational (a) and rotational components (b).
4 Application of TFRS of rotational motion TFRS can reflect the distribution of seismic re-
sponse with time and period. Because TFRS takes all characteristics of seismic motions into account, it is a useful tool for structural analysis. As an example, one twelve-storey frame concrete structure (Figure 9) is chosen to analyze its rotational damage mechanism by using the TFRS of the rotational motions.
The TFRS of the torsional component is chosen (see Figure 6a) to analyze the structure. When the peak value of the input torsional acceleration is 0.5 rad/s2, the
structure is initially in the elastic range, and about 34 s later the structure comes into the plastic deformation range and the stiffness of the structure degrades gradu-ally, its natural period is changed simultaneously with time (Figure 10). It illuminates that the intensity of the input ground motion will have influence on the fre-quency characteristics of the structure.
The change process by times of natural period is defined as the time history of natural period of the structure. The time history of natural period of the
Figure 9 The plane layout of one twelve-story structure. Figure 10 Time history of the basic natural period (T) of the twelve-story structure.
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twelve-storey structure and the contour diagram of TFRS of the torsion component are shown together in Figure 11. The figure indicates that the earthquake oc-curred about 34 s later, the structure is impacted by a series of large torsional components and it comes into the plastic range, accordingly, its basic natural period changes from 0.98 s to 1.27 s. At about 42 s on the time axis the structure meets the large torsional motion, of which dominant period is about 1.27 s. As mentioned above, when the vibration period of seismic motion is consistent with the changed natural period of the struc-ture, it is easy to produce resonance and destroy the structure. As a result, the twelve-story structure is de-
stroyed at about 42 s. The time history of the natural period of the struc-
ture is defined as damage line. It is obvious that each structure has its own damage line, so that only the seis-mic wave, which has suitable TFRS, can be used to make seismic analysis. And the other way round, the seismic design will not be reliable. As indicated by the arrow in Figure 11, if the amplitude spectrum of the torsional mo-tion is not so large at about 42 s on the time axis and at about the 1.2–1.3 s on the period axis, which is corre-lated with the damage line of the structure, the structure would not be destroyed by the torsional motion.
Figure 11 Time history (thick line) of natural period (T) of the structure and contour diagram of the time-frequency response spectrum of the torsional motion.
5 Discussion and conclusions Our research shows that the rotational ground mo-
tion of the Jiji (Chi-Chi) earthquake can be synthesized based on translational acceleration records from one sta-tion. But the dominant peak amplitude spectra, which are inferred from the data of Li-Yu-Tan Dam seismic array recorded in the Jiji (Chi-Chi) earthquake, are around 5 s for both torsional and rocking motions and another peak amplitude spectrum is found around 1.43 s for the tor-sional motion (Huang, 2003). It is different from what we got in this paper. The dominant peak amplitude we calculated is around 0.9 s for torsional motion (see Fig-ure 6) and around 0.45 s for rocking motion.
Also different from the results got by Huang (2003), Figures 5, 6 and 8 show that the torsional component is richer than the rocking component in period domain. It implies that one of the ground motion components can not reflect the characteristics of the seismic motions
completely and the epicentral distance and site condition are more important for us to study the ground motion characteristics. The characteristics of each component of ground motions, especially rotational motions, need to be further studied.
The duration of seismic motion is introduced into TFRS, for it can reflect the changes of amplitude of dif-ferent component with period and time simultaneously, especially the change in the frequency (or period) do-main characteristics of ground motion with time.
The TFRS can be used to discuss the cumulative effect of the seismic ground motions on the structure, and to analyze the damage mechanism of structure indeed.
The proposed definition of damage line of the structure is important for seismic design. Only when the input seismic wave has suitable TFRS, the structural seismic analysis and seismic design are reliable. The TFRS may provide help for our choosing the input seismic wave for seismic analysis.
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Acknowledgments The project was funded by the National Natural Science Foundation of China under grant No.50578125.
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