The response spectrum theory is one of the most important theories in seismic design. It contains several basic assumptions:(1) The structural seismic response is elastic and superposition principle can be used for modal combination.(2) The earthquake motions are identical at every structural support and soil-structure interaction is ignored. In addition,many given response spectrums were not computed by one-time seismic response and did not distinguish some influential factors. They were envelope curves of different seismic response spectrum called standard response spectrum or average response spectrum.(3) The critical seismic response of structures is the maximum response and it is irrelevant to other dynamic factors such as the times or probability around maximum response(Hu,2006).

Response spectrum is the maximum response value of a series single degree freedom system with the same unit mass,damping and different frequency under certain seismic motion. Actually,it is a regulation for design seismic force and not reflects a specific time-history feature. Being a quasi-static response theory,response spectrum theory has some limitations:(1)It can only reflect the maximum elastic response of structures,but not the inelastic seismic response of structures. However,the latter tallies better with the actual.(2)It cannot reflect the effect of low-cycle fatigue property and collapsing analysis under rare earthquake.(3)Calculation of response spectrum mainly depends on the most intense part of ground motions(Xie et al.,2003).

Seismic design by response spectrum theory simplifies dynamic problems to static problems. Based on superposition principle,we can get maximum response value of complex constructions under input ground motions. However,it only takes the amplitude and frequency characteristic of earthquake into account,without the duration characteristic. The impact to structural damage caused by duration characteristic of ground motion should not be ignored. The peak acceleration of American Parkfield earthquake record(1966) was 0.15g and that of American Stonecanyon earthquake record(1972) was 0.169g. The degree of damage around the stations in these two earthquakes were relatively slighter. That is because the duration of the two earthquakes was relatively short. Although the peak acceleration of Mexico earthquake record(1962) was only 0.105g,for its long duration,the damage around the station was serious. After studying the development of a reliable analytical model for seismic analysis of steel structures,Shen and Wu(2007)concluded that the damage cumulation effect was considerable and important in structural seismic analysis. Analyzing a suite of seven structural models with strength and stiffness degrading characteristics by a suite of almost 500 strong-motion accelerograms, Bommer et al.(2004)believed that seismic performance evaluation method for masonry structure should be improved and take duration into account. Duration has a close relationship with structural cumulative damage and low cycle fatigue effect(Du et al.,1992). According to soil condition and duration of strong ground motion, Sheng and Luo(2008)classified 302 Northridge in USA and 339 ChiChi in Taiwan earthquake ground motion records. Based on a great deal of nonlinear time history analysis of single degree of freedom with normalized peak ground acceleration and three hysteretic models,they found that mean hysteretic energy is determined by site conditions,duration of strong ground motion and damping.

Elastic-plastic time-history analysis takes elastic-plastic properties of structures into consideration. With structural stiffness increasingly changes,we can get displacement,velocity and acceleration time-history response of each particles under input motions by step by step calculation. Though this method has considered the seismic duration characteristic,it is time-consuming and its calculation is complex. Furthermore,the result precision depends on model element division,material constitutive relation and selection of motions. For there are lots of data processing,elastic-plastic time-history analysis has a high demand for hardware. For the reasons above, this method is only applied to important projects such as nuclear power station,large dams,large bridges and high-rise buildings. In addition,the input of seismic duration is critical for dynamic seismic analysis method. Whether using revised natural seismic wave or artificial wave,to speak properly,the anti-seismic design only reflects anti-seismic reliability of structures under seismic waves we have selected.

Duration characteristic of ground motion isn't taken into account in response spectrum theory. Dynamic analysis method can reflect amplitude,frequency and duration characteristic of earthquake. Even though seismic design of a structure by elastic-plastic time history analysis under certain input ground motions is safe, it may not quite safe under other input ground motions. In order to solve above-mentioned problem,Time- Frequency Response Spectrum was proposed(Zhang et al.,1999; Zhang,2000; Luo et al.,2004). In this paper, the Time-Frequency Response Spectrum is first introduced. Then,the Time-Frequency Response Spectrum of Tianjin record SN direction(Tianjin people's hospital recording station),ChiChi record SN direction(Nantou Hsinjie middle school recording station)and Wenchuan record EW direction(Wolong recording station)are analyzed. After that,the conception of Normalized Time-Frequency Response Spectrum(NTFRS)is proposed. The NTFRS of the three above-mentioned typical seismic waves are calculated and analyzed.

The formula of acceleration response spectrum is

(1)

Sa(ξ,T)is the value of acceleration response spectrum. ξ,T is damping ratio and natural period of SDOF system respectively. Time-Frequency Response Spectrum is the response value of a series single degree freedom system with the same unit mass,damping and different frequency under certain seismic motion(Zhang et al., 1999). It is a function of duration and frequency:

(2)

Although the definitions of the two spectrums above are different,the calculation methods of them are similar. Therefore,the new definition of spectrum is still called response spectrum,but added a Time-Frequency before it. There are three advantages of Time-Frequency Response Spectrum(TFRS)(Zhang et al.,2002). First,having some features of response spectrum,people who are familiar with response spectrum accept TFRS more easily. Second,we can do research by combining time-frequency feature of seismic motions with structural dynamic property and failure mechanism. Thirdly,being different from the three-dimensional spectrum proposed by Safak and Franke(1995),it directly includes duration and can be used to study the accumulated effect.

Knowing from the definition of TFRS,it is a three-dimensional spectrum which contains three features of earthquake motions(amplitude,frequency and duration). TFRS can reflect the variation of structural response with seismic duration and period simultaneously. Zhang(2000)calculated and analyzed the TFRS of American El Centro waves. Luo and Li(2004)proposed the standardized concept of TFRS and calculated standardized TFRS of American El Centro waves and so on. Then a 6-story frame was simplified to a SDOF elastic-plastic system and its restoring force curve was obtained by pushover analysis method. After that,they put forward the concept of structural damage curve. Guo et al.(2004)calculated the temporal frequency response spectra of Shidian MS 5.9 earthquake records observed in Taiping and some other sites and got temporal frequency response spectra figures for three directions. Huang(2006)analyzed the impact of damping on TFRS,proposed the concept of elastic-plastic TFRS and anatomized the influences of yielding displacement on it. The rotational seismic motions were estimated in terms of the records from the 2008 Wenchuan earthquake based on the theory of elastic half-space plane harmonic wave propagation. TFRS of the rotational motions were calculated and the change regularities of its response spectrum shape were analyzed in the time and frequency domain. Then,the torsional seismic motion was applied to analyze the failure mechanism of one reinforced concrete asymmetric frame structure(Che,2009; Che et al.,2010). Based on TFRS,the definition of accumulative kinetic energy curves was presented and together with modal participating mass ratio,the definition of potential damage energy(PDE)was proposed. The potential destructiveness for selected pulse-like and non-pulse-like ground motions of Northridge earthquake to a 12-storey RC framed structure was predicted quantitatively by PDE and the prediction results were verified by elastic-plastic time-history analysis(Qiu et al.,2014).

The TFRS of Tianjin SN record direction,ChiChi SN record direction and Wenchuan EW record direction were calculated(Fig. 1). For having too long durations,we intercepted 0ξ40 s parts of ChiChi SN record and 20ξ90 s part of Wenchuan EW record respectively as effective seismic duration. From Fig. 1,we can see the TFRS of three motions differ greatly. The TFRS of Tianjin SN record is like a trumpet and has one independent peak. Its TFRS is continuous along duration axis and has certain width along period axis. The TFRS of ChiChi SN record is zigzag and has lots of peaks. Its TFRS is continuous along duration axis and has certain width along period axis. The TFRS of Wenchuan EW record has three independent peaks along duration axis and the peaks become smaller gradually. Their different characteristics of TFRS will be discussed below.

Fig. 1

Fig. 1 Time-frequency Response Spectrum(Tianjin SN,ChiChi SN and Wenchuan EW seismic waves)

For the TFRS peak values of seismic waves are different and the scale of acceleration axis differs greatly, we used normalization method to make them comparable. The normalization method is as follows. Different seismic TFRS are divided by their maximum spectra value and the new spectrum is called Normalized Time- Frequency Response Spectrum(hereafter referred to as NTFRS). For convenience,the contour plots of Tianjin SN,ChiChi SN and Wenchuan EW seismic waves were drawn in Fig. 2 to 4.

Fig. 2

Fig. 2 Normalized Time-frequency Response Spectrum of Tianjin SN seismic wave

Fig. 3

Fig. 3 Normalized Time-frequency Response Spectrum of ChiChi SN seismic wave

Fig. 4

Fig. 4 Normalized Time-frequency Response Spectrum of Wenchuan EW seismic wave

Though the duration of Tianjin SN wave is only 19.19 s,the peak area(dark red parts)of its NTFRS is wide. Basically,it covers the area of duration 7.0ξ19.19 s and natural period 0ξ2.7 s. The peak area of Tianjin SN wave NTFRS has several similar-sized,nearly parallel,compact primary and secondary peak ridges. As one can imagine,before natural period 7.0 s,the structural responses of different natural periods are small. But,starting from 7.0 s,structures of natural periods between 0 s and 1.9 s will suffer the impact from primary peak ridge of NTFRS. Even if structures may not collapse,they easily get plastic deformations and the natural period is increased. Structures with increased natural period will sequentially suffer the impact from secondary and tertiary peak ridge of NTFRS along duration axis and be damaged severely. At last,only a small impact may lead to a totally collapse(Fig. 2).

The intercepted duration of ChiChi SN wave is up to 40.0 s,but the peak area(dark red parts)of its NTFRS is narrow. Basically,it covers the area of duration 6.0ξ40.0 s and natural period 0ξ0.8 s. The secondary peak ridge of its NTFRS is beyond 1.9 s between duration 12.0 s and 16.0 s. As one can imagine,when suffering moderate intensity ground motions,structures of natural periods between 0 s and 1.9 s will probably enter the plastic state and then suffer cumulative damage. However,structures of natural periods between 0.8 s and 7.0 s will only suffer the impact from secondary and below secondary peak ridges of NTFRS. Even if structures are slightly damaged,it will not suffer cumulative damage for enter into subsequent,small peak ridge area(Fig. 3).

The intercepted duration of Wenchuan EW wave is up to 70.0 s. The peak area(dark red parts)of its NTFRS can be divided into three parts: part one between duration 8.0~16.0 s and natural period 0~0.8 s,part two between duration 29.0~38.0 s and natural period 0~0.7 s,part three between duration 64.5~65.5 s and natural period 0~0.3 s. As one can imagine,when suffering moderate intensity ground motions,structures of natural periods between 0 s and 0.7 s will probably enter the plastic state and then suffer cumulative damage. However,structures of natural periods between 0.7 s and 7.0 s will only suffer the impact from tertiary and below tertiary peak ridges of NTFRS. It will not be damaged or may suffer slight damage(Fig. 4).

The response spectrum theory can not reflect the inelastic seismic response of structures and structural performance change with time in real earthquake. When structures reach the plastic stage in a severe earthquake, the duration characteristic of ground motion will be more important. Luo and Li(2004),Li S D(2004)defined the natural period variation curve of structural response from elastic stage to plastic stage under earthquake as the structural damage curve. Qiu et al.(2013)comparatively analyzed the structural damage mechanism of a 10-storey RC frame structure under two seismic waves whose time-frequency characteristics were totally different by structural damage curve.

Due to space limitations,we selected ChiChi SN seismic wave(Fig. 3)and adjusted its PGA to 500 Gal (equivalent to intensity 8 rare earthquake level). Then,a 12-storey reinforced concrete frame structure subject to the adjusted ChiChi SN seismic wave is studied by elastic-plastic time-history analysis method. Both the front and side of this building are symmetrical three-bay and the span of side-span and mid-span is 5.8 m and 3.4 m,respectively. The story heights are all 3.2 m,except the ground floor is 4.2 m. The size of column crosssection and beam cross-section is 550 mm×550 mm and 300 mm×600 mm,respectively. The plate thickness is 120 mm and the strength classes of concrete of all components are C30. The compressive strength is fc=30.0 MPa,mass density is ρ=2.4 t·m⁻³,modulus of elasticity is E₀=30.0 GPa,Poison's ratio is v=0.2 and shear modulus is G=12.5 GPa. The distributed steel is HRB335 and its tensile strength f_y=300 MPa,mass density ρ=77 t·m⁻³ and modulus of elasticity is E₀=200.0 GPa.

The acceleration time history of structure-top corner is showed in Fig. 5. The maximum acceleration response value is –5.26 m·s⁻² at 12.1 s. The structure didn't collapse at that time. However,at 13.2 s when the acceleration response value is –3.68 m·s⁻² that is 70% maximum acceleration response value,the structure collapses for serious damage. This demonstrated when structure response entered into the plastic stage,seismic design by maximum response value has some limitations.

Fig. 5

Fig. 5 The acceleration time history of structure-top corner

In order to get structural damage curve,this paper used the method in reference(Qiu Z G and Luo Q F,2013). According to development situations of plastic hinges of frame beam column nodes at different time,the equivalent spring stiffness at corresponding positions was modified. Then,we can get the changing curve of structural natural period. The development situations of structural plastic hinges at 7.5 s,9.0 s,12.0 s and 13.2 s are showed in Fig. 6. Pink means hinge yields. Dark blue means the ability level of hinge is direct use. Light blue means the ability level of hinge is life safety. Yellow(green) means the ability level of hinge is collapse prevention and the hinges begin to lose bearing capacity.

Fig. 6

Fig. 6 The development of structure plastic hinge

Structural damage curve and NTFRS of ChiChi SN seismic wave were plotted together in Fig. 7. Structural natural period fluctuates greatly from 1.41 s to 2.21 s between duration 7.3 s and 10.1 s. However,structural damage curve is in the smaller peak area of ChiChi SN NTFRS. This explains that energy dissipation components will enter into the plastic stage under reciprocating action by small ground motions during the beginning phase of severe earthquake. Therefore,structural natural periods in elastic stage are different from that in the peak area of motions.

Fig. 7

Fig. 7 Structural damage curve

Structural damage curve reached the peak area of NTFRS at 12.1 s for the first time. Structural natural period rose slightly. But,when structural damage curve reached the second peak area of NTFRS at 13.2 s, structural natural period rose to 3.89 s. After that,the structure collapsed at a smaller peak area of NTFRS. The analysis result indicates that structural damage curve together with NTFRS can be used to analyze structural failure mechanism.

The impact to structural damage caused by duration characteristic of ground motion should not be ignored. Time-Frequency Response Spectrum(TFRS)is a three-dimensional spectrum,which includes three main characteristics of ground motion: amplitude,frequency characteristic and duration. TFRS can reflect the variation of structural response with seismic duration and period simultaneously.

Based on TFRS,Normalized Time-Frequency Response Spectrum(NTFRS)was proposed. The NTFRS of three typical seismic waves were calculated. By comparison,we found that the TFRS peak values of seismic waves are different and the scale of acceleration axis differs greatly. Earthquake waves with different NTFRS may cause different effect on the structures. Therefore,seismic design should take suitable ground motions for certain structures,or the seismic design results may be less reliable.

One 12-storey reinforced concrete frame structure was analyzed by using elastic-plastic time-history analysis method. Under strong motions,the natural vibration period of structures would increase step by step. By comparison between seismic response of structures,structural damage curve and NTFRS of input motions, we found that the maximum seismic response of structure might not be the cause of its collapse and seismic design based on maximum elastic response of structures had some limitations. It could be concluded that the Normalized Time-Frequency Response Spectrum is better for analyzing the characteristics of ground motions and the structural failure mechanism in the earthquakes. It is worth to further study the NTFRS.

This work was supported by the Kunming University School-Level Program(XJL15007),Kunming University Talent Introduction Plan(YJL15001).