A Normalized Hysteretic Energy Spectrum for Energy-Based Seismic Design

. To disclose the impacts of hysteretic energy (HE) demand on energy-based seismic design, this paper introduces the dimensionless parameter EH (cid:2) to express the cumulative HE indirectly and establishes the EH (cid:2) spectrum for energy-based seismic design. After analyzing numerous seismic responses of a single degree-of-freedom (SDOF) system, the author set up a simplified EH (cid:2) spectral formula based on the genetic algorithm. Then, 750 ground motion records were selected according to Chinese site classification, and used to examine the effects of soil type and damping ratio on the EH (cid:2) spectra. The results show that the soil type, site group and damping ratio have significant effects on the EH (cid:2) spectra; the ductility ratio has an impact on the spectral value but not the spectral shape.

The hysteretic energy (HE) demand is the key to the energy-based seismic design, owing to its relevance to the cumulative structural damage induced by seismic activity. Considering its simplicity, convenience and applicability, scholars at home and abroad have proposed various forms of HE spectra. For instance, Mckevitt et al. [12] analyzed the HE of multi-story buildings under seismic excitation, revealing that most HE is dissipated from the bottom floor of the structure under the uniform distribution of stiffness and strength along the structural height. Khashaee P. [13] established an HE spectrum in light of the field effects and ground motion features, such as severity, duration and frequency. Through linear and nonlinear dynamic analysis, López-Almansa et al. [14] derived the HE spectrum of equivalent velocity ratio from the record of strong earthquakes in Turkey, while considering the impacts of soil type and earthquake magnitude. On 89 pairs of bidirectional seismic motion records, Wang et al. [15,16] established the mean normalized input energy spectra and HE spectra, and created a normalized HE spectrum of constant ductility ratios to estimate the story HE demand, where the normalized HE is defined as the ratio of the HE to the square of the peak ground acceleration (PGA). Sun et al. [17] defined the ratio of the equivalent velocity of HE to the peak ground velocity (PGV) as a dimensionless parameter βEh for indirect expression of the HE, developed the βEh spectra against the regression results (e.g. seismic impact, soil type, damping ratio and ductility) on various seismic responses of the single degree-of-freedom (SDOF) system, and set up the relationship between the PGV and the PGA. Dogru et al.
[18] assessed the energy parameters against the total energy input and HE for special steel concentrically braced frames (CBFs) with different heights, conducted nonlinear dynamic time history analysis on the HE variation along the frame height, and eventually derived the seismic energy demand spectrum and HE distributions of the CBFs.
In general, none of the existing HE spectra refers to Chinese site classifications. The Chinese codes divide the building sites into five classes, and further split each class into three groups, according to the predominant period of ground motion. This paper selected 750 ground motion records by Chinese site classification, and derived the simplified βEH spectra of cumulative HE demand using the energy-balance equation of SDOF system. In addition, the author examined the effects of soil type, structural damping ratio and ductility ratio over the HE spectra, and presented the mathematical expression of simplified βEH spectrum.

Energy-balance equation
Bruneau and Wang [19] advised to calculate the seismic input energy by the relative energy equation.Under unidirectional horizontal ground motion,the relative motion equation of an elastic-plastic SDOF system can be written as: where m is the mass; c is the viscous damping coefficient; fs is the restoring force; x, is the ground acceleration.
The energy equation can be derived from equation (1) through integration over the entire seismic duration: where t is time.
Equation (2) can be rewritten as: where Ekr is the kinetic energy; D E is the energy dissipated by viscous damping; EE is the elastic strain energy stored in the SDOF system; EH is the HE dissipated from inelastic behaviors; EIr is the total input energy induced by the earthquake.
In far-field earthquakes, structural failure is mainly caused by the cumulative damage from the cyclic effect and gradual accumulation of oscillation-induced seismic energy. Thus, the cumulative HE can reasonably describe the far-field seismic damage. Then, the EH can be expressed as an equivalent velocity VEH [20]: where m is the mass.Finally, the dimensionless parameter βEH can be defined as the ratio of the equivalent velocity of HE to the PGV [17]:

Spectral parameters
The selected SDOF system satisfies the forcedisplacement relationship of the bilinear elastic-plastic model. The post-yielding stiffness ratio (PYSR) was set to 0.05 and 0.00, the damping ratio ζ to 0.01~0.20, and the ductility ratio μ to 1~10.

Ground Motion Records
A total of 750 ground motion records were extracted from the PEER Ground Motion Database according to the geological conditions of various seismic stations [21] and the Code for Seismic Design of Buildings [22]. The number and percentage of ground motion records in each site type are respectively presented in Table 1 and Table  2.
In Chinese codes,according to the equivalent shear wave velocity of the soil layer and the thickness of the site cover,the building sites can be divided into soil type I,Ⅱ,Ⅲ,Ⅳ.It should be mentioned that Lv [23],by analyzing a number of geological prospecting data of U.S. station sites, concluded that China's soil type I corresponds to the site class A and B and a part of site class C of U.S., China's soil type II is between site class C and D of U.S., China's soil type III is between site class D and E of U.S., and China's soil type IV is identical with U.S. site class E.  [24], where aE and vE are 1/4 of the platform values corresponding to the absolute acceleration response spectrum and pseudovelocity response spectrum, respectively. The damping ratios of both spectra are 0.05.
According the Seismic Ground Motion Parameter Zonation Map of China [25], the ground motion records of each soil type (I, II, III and IV) in Table 3 were further divided into three site groups according to the Tg.

HE Spectra
The cumulative HE spectra are determined by the features of the estimated seismic oscillation at a given site and the dynamic behavior of the structure. This section explores the impacts of soil type, site group, structural damping ratio and ductility ratio on the HE demand of the SDOF system, which is represented by the dimensionless parameter EH E .

Soil type
The mean EH E under the ground motions of site group 1 of soil types I, II, III and IV (Table 3) was computed at the ductility ratio of 2, the damping ratio of 0.05 and the PSYR of 0.00. It can be seen from Figure 1 that the soil type had a significant impact on the HE. The mean EH E spectra consist of the rising, stable and declining segments, which correspond to the short, medium and long periods, respectively. The EH E spectra of soil types I and II were relatively stable in the long term, but that of soil type IV plunged with the increase of the period.
The EH E spectrum of soil type III fell between those of soil types I and II and soil type IV. From soil type I to IV, the peak EH E and peak period increased continuously.

Structural damping ratio
The mean EH E spectra of the site groups 1, 2 and 3 in the soil type II at different damping ratios are displayed in Figure 2  As shown in Figure 2, the spectra underwent a negligible shift towards the right with the growing damping ratio. Meanwhile, the peak EH E of each site group dropped gradually, revealing the peak clipping effect of the damping ratio. The damping ratio had similar impacts on the EH E spectra, under the ground

Ductility ratio
The mean EH E spectra of the site groups 1, 2 and 3 in the soil type II at different ductility ratios are shown in Figure 3 (PGA=0.2g, ζ=0.05, p=0.0 and μ=2, 3, 4, 5, 6, 7 and 8). Obviously, the EH E spectra values are sensitive to small variations in ductility ratio when the damping ratio remains constant. As the ductility ratio increased from 2 to 4, the EH E value at constant period grew continuously, but became stable when the ductility ratio reached and surpassed 5. The phenomena demonstrate the limited effect of ductility ratio on spectral shape.

Site group
The mean EH E spectra of the site groups 1, 2 and 3 in the soil type II are displayed in Figure 4 at PGA=0.2g, μ=2, p=0.05 and ζ=0.05. It can be seen that the site group has an important impact on the HE for the same soil type. The EH E spectra values increased linearly in the short period when the site group changed from 1 to 3.

Simplified βEH Spectra of Cumulative HE
As mentioned above, the EH E spectra consist of the rising, stable and declining segments, and the spectral values are affected by the soil type, site group, damping ratio and ductility ratio. Hence, 12 groups of ground motions were classified by soil type and site group, and taken as the inputs. Then, the mean response of each group of ground motions was computed for the statistical analysis on the impacts from the damping ratio and ductility ratio. Figure 5 provides the fitted smooth spectral curves. The corresponding mathematical expressions are as follows: T is the separation period between the rising segment and horizontal segment, and 2 T is the separation period between the horizontal segment and declining segment; (1) Correction factor of damping ratio ( 1 K ) The reference damping ratio of the elastic-plastic system was set to 0.05. Considering the peak clipping effect of damping ratio on the EH E spectra of cumulative HE, the correction factor can be defined as: By analyzing the EH E spectra peak values under different structure damping ratios, it is found that the peak value decreases continuously with the increase of structural damping ratio. Such decrease is unrelated to soil types and site groups. Therefore, 1 K can be fitted by an inverse proportional function.The comparison between EH E peak values at different damping ratios shows that the peak value decreased continuously with the increase of the damping ratio. However, the decrease has nothing to do with soil type or site group. Hence, the 1 K can be fitted by an inverse proportional function: (8) where ζ is the damping ratio.
(2) Correction factor of ductility ratio ( where μ is the ductility ratio. (3) Attenuation index ( J ) of the declining segment As shown in Figure 2, the declining segment tended to be stable with the increase of damping ratio.
where the values of 1 J are listed in Table 4, which are related to soil type and site group; ζ is the damping ratio.

T T ) of characteristic points
The periods 1 T and 2 T , corresponding to the starting and end points of the horizontal segment, are related to soil type, site group and structural ductility, but their correlations with damping ratio can be neglected.There's a certain linear relationship between 1 T (or 2 T ) and μ: For the sake of simplification, the effects of μ can be neglected and the 1 T and 2 T values are listed in Table 4.
As shown in Table 4, max , EH E is the peak EH E in the stable segment at the acceleration amplitude of 0.2g. Here, the mean EH E values of each site group and soil type were selected, and fitted by the genetic algorithm on the Matlab. Figure 6 presents the fitted EH E spectra and the dynamic analysis results at different parameters.

Conclusions
Considering the importance of structure energy demand in energy-based seismic design, this paper establishes normalized cumulative HE spectra according to Chinese soil site classifications, and draws the following conclusions through in-depth analysis: (1) The soil type, site group and damping ratio have significant effects on the EH E spectra. As the soil type changed from I to IV, both the peak EH E and characteristic period increased continuously. The damping ratio has a peak clipping effect on the EH E spectra, and the effect remains the same in different site groups. For the same soil type, the peak EH E increased significantly as the site group changed from 1 to 3.
(2) The shapes of EH E spectra have nothing to do with ductility ratio, while the spectra values are positively correlated with structural ductility when P ≤4 and remains stable when P ≥5.
(3) The EH E spectra consist of a rising segment, a stable segment and a declining segment. The separation periods 1

T and 2
T are related to soil type, site group and ductility ratio, but not to damping ratio.