Damage-based yield point spectra for sequence-type ground motions

Abstract

Structural damage caused by mainshock can further be aggravated by aftershocks, which can lead to structural collapse. The current practices on the seismic design of structures generally only consider mainshock effects. This manuscript therefore presents the investigation on damage-based yield point spectra (YPS) of a single-degree-of-freedom (SDOF) system under sequential earthquakes. The collected sequence-type ground motions are recorded from 16 earthquake events and classified to four classes. The aftershocks in sequence are scaled according to different relative intensity levels. The modified Park-Ang model, which consists of maximum displacement and hysteretic energy dissipation, is employed to calculate YPS. The effects of period, ductility factor, damage index, site category, aftershock intensity, and structural damping are statistically studied. The results prove that the strong aftershock ground motion has more distinct influences on the YPS. In particular, the yield strength coefficient demand under seismic sequence increases by 10%–50%. The yield strength coefficient demand determined by the damage-based YPS is greater than that determined by the ductility-based YPS—the former is 10%–40% higher than the latter. Finally, the empirical expression of damage-based YPS is established by statistical mean method and regression analysis.

1 Introduction

Since the 1990s, several major earthquake disasters have made the world’s earthquake engineering community keenly aware that under strong earthquakes, particularly those that exceed the design strength of structures, can severely damage buildings designed with the force-based method. As a result, grave economic losses have been sustained, and the seismic design concept that is based solely on ensuring life safety cannot meet the needs of social development. Compared with the force-based design, the displacement-based design focuses on displacement as the design target. Displacements can better reflect the nonlinear response of structures that experience strong earthquakes and can well control the functional state of structures during earthquakes. The displacement-based seismic design method has therefore been extensively developed (Moehle 1992; Kowalsky et al. 1995; Chopra and Goel 2001; Rossetto and Elnashai 2005; Powell 2008; Kowalsky et al. 2010). Currently, the widely used seismic design approach, which can achieve displacement-based structures, is the modified capacity spectrum method proposed by Chopra and Goel (1999), and Fajfar (2000). In this method, first, the capacity curve of structures is obtained by nonlinear static analysis, and the multi-degree-of-freedom system is transformed into an equivalent single-degree-of-freedom (SDOF) system. Second, the acceleration–displacement response spectrum, which uses spectral acceleration and spectral displacement as vertical and horizontal coordinates on the same graph, represents the capacity and displacement demands, respectively. The seismic demand spectrum can thereafter be obtained by reducing the elastic spectrum using a strength reduction factor. Finally, the system displacement response is solved by the intersection of the capacity and demand curves. Aschheim and Black (2000), Aschheim (2002), Tjhin et al. (2007) analyzed axially loaded members, cantilever shear wall subjected to lateral forces, multi-layer shear wall structure, and bending frame. It was found that the yield displacement (Δ_y) of the structure is a relatively stable parameter. The yield point spectra (YPS), a variant of the capacity spectrum method, are thus proposed. In lieu of period, the relatively stable yield displacement is used as the initial design parameter in the YPS method. The method determines the strength demand of structures by directly controlling the structural ductility demand or peak displacement response that achieves various expected performance levels. Some researchers (Safar and Ghobarah 2008; Kotsoglou and Pantazopoulou 2009) used the YPS method to analyze the maximum impact of structures during earthquakes; the results have fewer errors than the dynamic time history analysis results. The YPS can be used not only for structural performance evaluation, but also for structural design. For example, Tabatabaei and Rahmanian (2012) designed a frame structure with the YPS method.

After the mainshock causes damage to the structure, the aftershocks usually further increase the degree of structure damage. The compounding effect of damage and disruption caused by seismic sequence results in tremendous losses to society, as exemplified by the aftermaths in Wenchuan (2008), Wang (2008), Christchurch (2010–2011), Moon et al. (2014), Tohoku (2011), Kazama and Noda (2012), and Nepal (2015), Chen et al. (2017). Understanding the sequence-type ground motion and its impact on structure response is therefore crucial to the improvement of structure resilience.

Currently, researchers have investigated the influence of aftershocks on structural damage in different ways. Some researchers focused on the effects of seismic sequence on inelastic spectra for structural design, such as the strength reduction factor R spectra (Hatzigeorgiou 2010a; Zhai et al. 2015; Sun et al. 2016), structural damage D spectra (Zhai et al. 2013), and ductility factor μ spectra (Hatzigeorgiou 2010b; Goda and Taylor 2012). In addition, several investigations studied changes in structural response, e.g., response of reinforced concrete structures (Raghunandan et al. 2015; Efraimiadou et al. 2013; Shen et al. 2019) steel frame buildings (Li and Ellingwood 2007; Ruiz-García and Negrete-Manriquez 2011), and wood frame building (Goda and Salami 2014; Nazari et al. 2015), subjected to sequence-type ground motions. The results in the above investigations prove that larger maximum displacements or more severe structural damage because of seismic sequence compared to those caused by mainshock. Therefore, the influence of aftershock should be considered in the structural design stage. Unfortunately, most current seismic design codes around the world are designed for single earthquake without considering the influence of aftershock.

As previously mentioned, aftershocks aggravate structural damage. The current capacity spectrum and YPS, however, do not reflect the effect of aftershock. Therefore, this manuscript studies the damage-based YPS through numerical analysis of serial nonlinear SDOF systems subjected to sequence-type ground motions. The aftershock ground motions are scaled to have different intensity levels. The influences of period, ductility factor, damage index, site category, aftershock intensity, and structural damping on the YPS are statistically studied. Finally, a predictive model of damage-based YPS is established by statistical mean method and regression analysis.

2 Selected sequence-type ground motions

The actual sequence-type ground motion may contain one or more aftershock ground motions. Studies have shown that different numbers of aftershock ground motions increase the structural damage to varying degrees, and the cumulative damage effect may be more severe with the increase in number of aftershock ground motion (Zhai et al. 2013; Goda and Taylor 2012; Li and Ellingwood 2007). In order to facilitate the comparative analysis, the sequence-type ground motion selected in this manuscript is composed of an aftershock ground motion and a corresponding mainshock ground motion.

Researchers have conducted related studies on the sequence-type ground motion. The available ground motion data for research, however, is insufficient because of limited seismic data. Most of these studies thus use artificial ground motion (Raghunandan et al. 2015) or repetitive ground motion (Hatzigeorgiou and Beskos 2009; Hatzigeorgiou and Liolios 2010), but the use of these ground motions may result in significant overestimations of structural drift demands (Ruiz-García and Negrete-Manriquez 2011). The recorded sequence-type ground motions used in this study are therefore selected from the Pacific Earthquake Engineering Research Center (PEER) Next Generation Attenuation (NGA) relationships database (Pacific Earthquake Engineering Research Centre 2018) and strong-motion seismograph networks (K-NET, KiK-net) (National Research Institute for Earth Science and Disaster Resilience 2018). The selection principles are as follows.

• (1) The earthquake magnitudes of mainshock and aftershock are not less than 6.0 and 5.0, respectively, excluding earthquakes that are less probable to cause severe damage to the building structure.

• (2) The fault distance is greater than 10 km to reduce the influence of near-field effect.

• (3) All records should originate from the same station and the same seismic event. The peak ground acceleration (PGA) of mainshock is larger than 0.10 g, while the PGA of aftershock is larger than 0.05 g.

• (4) To ensure that the structure is at rest before the aftershocks occur, a 100 s time interval is added between the selected mainshock and aftershock ground motions.

• (5) To study the influence of site class on the YPS, ground motions are classified using the site classification method of the United States Geological Survey.

Based on the selection principle, 342 recorded sequence-type ground motions (154 for site class B and 188 for site class C) are chosen from 16 earthquakes, as summarized in Table 1. The number of seismic sequences that meet the selection principle is very small on the site class A and D, and unable to perform relevant statistical analysis. To facilitate statistical analysis, the PGA of mainshock in all selected seismic sequences is adjusted to 0.2 g.

Table 1 Information of collected sequence-type ground motions

The relative intensity of aftershock ground motion (γ) is defined as

$$\gamma { = }\frac{{PGA_{as} }}{{PGA_{ms} }}$$

(1)

where PGA_as and PGA_ms are the PGA of the aftershock and the mainshock ground motion, respectively. Parameter γ represents the ratio of PGA_as to PGA_ms. The intensity of the former is generally lower than that of the latter because of the aftershock’s lower magnitude. In seismic sequences in the past, however, aftershock with intensities greater than those of mainshock exist. To investigate the influence of aftershocks on YPS therefore, this manuscript adopts 5 levels of γ: γ = 0.5, 0.8, 1.0, 1.2, and 1.5. In order to provide a design tool that considers different intensity of aftershock ground motions that exist in recorded sequence-type earthquake, the value of γ (i.e., 1.5) is used to simulate the extreme case of aftershock.

3 Structural damage and performance level

Maximum displacement is common indicator used in performance-based seismic assessment methods, such as performance-based frameworks employed in FEMA-356. The structural damage caused by earthquakes, however, has various forms. The degree of structural damage cannot be fully presented through maximum displacement only. Reasonable indicators must therefore be used to evaluate the degree of structural damage. The structural seismic response is related to the characteristics of the ground motion (amplitude, duration, spectrum, etc.), aftershocks may greatly increase the duration of ground motions and may cause low-cycle fatigue damage to structures, and this situation can be considered by the hysteretic energy. At present, it is generally believed that the maximum deformation and hysteretic energy of the structure are the main factors of structural damage. Two-parameter damage models are accordingly proposed. The widely known Park and Ang damage model, which was initially defined by Park and Ang (1985) and modified later by Kunnath et al. (1992), is employed in the present study. This damage model consists of a linear combination of normalized maximum displacement and hysteretic energy dissipation. The damage index D is expressed as

$$D{ = }\frac{{\mu_{m} - 1}}{{\mu_{u} - 1}}{ + }\beta \frac{{E_{h} }}{{F_{y} \mu_{u} x_{y} }}$$

(2)

where μ_m is the maximum ductility factor under earthquake ground motion, μ_u is the ultimate ductility capacity under monotonic loading, E_h is the cumulative hysteretic energy dissipation under earthquake ground motion, F_y and x_y is the yield strength and yield displacement, respectively. β is the energy dissipation factor and represents the rate of accumulated damage through hysteretic energy induced by cyclic loading, the value of β is 0.1 to represent frame structures (Negro 1997; Lu and Wei 2008) in this manuscript.

To associate the structural performance levels and damage index range, the structural performance level and damage index of the modified Park-Ang model should first be determined. In terms of structural performance level, several versions of performance definition exist. Five performance levels with descriptive damage extent and proposed damage index limits are adopted, namely Operational, Immediate Occupancy, Damage Control, Life Safety, and Collapse Prevention (Zhang et al. 2017). It is generally demonstrated that D = (0.4–0.5) is the repairable and unrepairable damage boundary proposed by the modified Park-Ang model, whereas D that approaches 1.0 represents total collapse. By associating the damage levels with the available calibration results of the modified Park–Ang model, the range of the damage index for each performance level may be obtained as summarized in Table 2.

Table 2 Relationship among the performance levels, damage index ranges and degree of damage (Zhang et al. 2017)

4 Construction procedure of yield point spectra (YPS)

4.1 YPS descripion

The YPS can be used to determine the strength demand to limit the peak ductility factor and drift responses of the structure. The yield point spectra plot the yield point of a single-degree-of-freedom system that has a constant displacement ductility factor (μ) for a range of periods on the axes of yield strength coefficient (C_y) and yield displacement (Δ_y). The yield strength coefficient is the ratio of the yield strength of the system (V_y) to its weight (W), and the ductility factor (μ_u) is the ratio of the peak displacement of the system (Δ_u) to its yield displacement.

Figure 1 plots the values of C_y versus Δ_y for the classic 1940 NS El Centro ground motion for μ_u = 1, 2, 3, 4, 6, 8, and 10. In this case, an ideal elastic–plastic load–deformation relationship is used; viscous damping is 5% of critical damping. When μ_u is 1, the YPS curve is the same as the capacity spectra curve. When μ_u has values higher than 1, the curves differ because the YPS and capacity spectra plot Δ_y and Δ_u, respectively. For the YPS, periods are constant along the radial lines extending from the origin.

Fig. 1

Yield point spectra calculated for the 1940 NS El centro record

4.2 Determination of damage-based YPS

Depending on the information available, an exact or approximate method can be used to determine the YPS. If a ground motion time series is available, the strength demand that corresponds to various ductility demands can be exactly determined, as shown by the curves in Fig. 1. Alternatively, if only the elastic response spectrum is available, then the strengths for the specified ductility demands can be approximately determined conventionally using a smooth R–μ–T relationship. This follows the same idea expressed by Chopra and Goel (1999) and Fajfar (2000) with the improvement of the seminal capacity spectra method (Freeman 1978), from which all these methods are derived. Considering the impact of cumulative damage on strength demand, the R–μ–D–T relationship is employed to determine the strength that corresponds to various ductilities (Zhang et al. 2017). In the calculations, the yield displacement corresponding to a given period and strength is determined using a simple relationship:

$$\Delta_{{\text{y}}} { = }\left( {\frac{T}{2\pi }} \right)^{2} S_{a}$$

(3)

where T is the natural period of the SDOF system; S_a is the pseudo-acceleration, given equivalently by C_yg; g is the acceleration of gravity.

In the present study, considering the impact of cumulative damage, the average yield point spectra are directly calculated because of the lack of elastic response spectra under seismic sequences. The proposed procedure for constructing damage-based YPS under the sequence-type ground motions that are illustrated in Fig. 2 can be summarized by the following steps.

• (1) Select the damping ratio (ζ) of the SDOF system and the value of β.

• (2) Define the target damage level (D_i) and ductility capacity (μ_u,j).

• (3) Select the natural period(T), calculate the initial stiffness of the system(K₀).

• (4) Determine the sequence-type ground motions, the elastic strength demand (F_e) is calculated by time history analysis.

• (5) Select the strength (ΔF). The yield strength (F_y) and yield displacement (Δ_y) of the specified system can be calculated as F_y = F_e − ΔF and Δ_y = F_y/K₀, respectively.

• (6) Calculate the peak displacement (Δ_u) and E_h through elastic time history analysis. The damage index (D) can be calculated according to Eq. (2). The analysis is repeated for sufficient values of ΔF to develop D that includes the target damage level (D_i) range of interest.

• (7) Change the value of T, and repeat steps (3)–(6) to determine the spectrum that is valid for the range of T.

• (8) Repeat steps (2)–(7) for different values of damage level and ductility factor.

Fig. 2

Flowchart of YPS computation

A series of SDOF systems are adopted to calculate the damage-based YPS under earthquakes. The elastic–perfectly plastic (EPP) model is employed due to its simple constitutive relationship. The SDOF systems with a set of 60 periods varies from 0.1 to 6.0 s with an interval of 0.1 s are considered. Five ductility factors (i.e., μ_u = 2, 4, 6, 8, and 10) and damage indices (i.e., D = 0.1, 0.2, 0.5, 0.8, and 1.0) are adopted to analyze different ductility performances and damage levels, respectively. The structural viscous damping ratio is assumed to be 5%.

5 Generation of damage-based YPS

Based on the selected 342 pairs ground motions, a total of 1 026 000 working stations are employed to obtain the average YPS with 60 periods of the SDOF system, five levels of ductility factor and damage index. The results are statistically analyzed on the basis of the period, ductility factor, damage index, site category, aftershock intensity, and structural damping. Due to the limited space, only the partial results are shown in the following sections, and other cases that have similar results are not discussed. It can be observed in Eq. 3 that Δ_y is proportional to C_y. The following studies therefore mainly analyze C_y in the study of the YPS.

The calculated YPS of the SDOF systems with different μ_u and D values under mainshocks and sequence-type ground motions with γ = 0.5 are shown in Figs. 3, 4, 5, 6. In order to analyze the influence of ground motion parameters and structural dynamic characteristics on the YPS, the YPS and C_y spectra are shown in Figs. 3, 4, 5, 6a and b, respectively. It can be observed that the YPS shows the same trend which is not affected by the ductility factor, damage index, site class, and type of ground motions. In the short period region (0–0.4 s), mean C_y increases sharply with increasing period. In the medium period region (0.4–2.0 s), C_y decreases significantly with the increase in period. The variation is gradual in the long period region (2.0–6.0 s), and C_y decreases with the increasing period of systems.

Fig. 3

YPS on site class B, D = 1.0, γ = 0.5: aC_y spectra, b YPS

Fig. 4

YPS on site class B, μ_u = 6, γ = 0.5: aC_y spectra, b YPS

Fig. 5

YPS on site class C, D = 1.0, γ = 0.5: aC_y spectra, b YPS

Fig. 6

YPS on site class C, μ_u = 6, γ = 0.5: aC_y spectra, b YPS

In the entire period region, C_y decreases with increasing μ_u, that is, the structural strength demand with high ductility factor is lower than that with low ductility factor when structures are subjected to seismic sequences. This indicates that the structure with adequate ductility can effectively resist earthquakes with a certain intensity. The ductility factor has a significant effect on C_y. Consider C_y on site class B with D = 1.0 as example. When structures subjected to sequence-type ground motions with γ = 0.5, the strength demand of the structure with μ_u = 2 is 1.49 times the strength demand when μ_u = 4 and 1.85 times when μ_u = 6.

For a given μ_u, C_y decreases with the increase in D. The structural strength demand with a high damage index is lower than that with a low damage index under the same conditions. This demonstrates that the damage in the high-strength structure is less than that in the low-strength structure under sequence-type ground motions. Damage index has significant influence on structural strength demand. For example, under the condition of site class B and μ_u = 6, the strength demand of the structure with D = 0.1 is 1.25 times of that of the structure with D = 0.2, and 1.85 times of the strength demand of the structure with D = 0.5 when subjected to sequence-type ground motion with γ = 0.5.

The yield point spectra reflect the relationship between Δ_y and C_y. The coefficients of variation (COV) of Δ_y and C_y spectra can therefore indirectly reflect the discrete form of YPS. It can be observed from Eq. (3) that the COVs of Δ_y and C_y spectra are the same. To study the extent of dispersion of YPS, the COVs of the corresponding C_y spectra are computed. The COVs of C_y under sequence-type ground motions with γ = 0.5 are shown in Fig. 7.

Fig. 7

COVs of C_y spectra, γ = 0.5: a COV on site class B, D = 1.0, b COV on site class B, μ_u = 6, c COV on site class C, D = 1.0, d COV on site class C, μ_u = 6

The COV increases with the increase in period, and in the relationship between the COV and ductility factor, the damage index is not evident. To some extent, it reflects the stochastic characteristics of seismic ground motion.

6 Effects of various parameters

6.1 Effect of aftershocks

The effect of aftershock on C_y spectra of YPS is discussed and the computed values of C_y of sequence-type ground motions and corresponding mainshocks are presented in this section. Figure 8 shows that that SDOF system’s C_y value increases with the increase in γ. Simply put, the stronger the aftershocks in the seismic sequence, the greater the C_y value necessary for the system. For structures damaged after the mainshock, subsequent aftershocks may cause additional damage as a result of cumulative damage. In order to satisfy a given ductility factor and damage index, the structure under a seismic sequence requires a greater yield strength than that subjected only to mainshock.

Fig. 8

YPS with different γ values, μ_u = 6, D = 1.0: a site class B, b site class C

In order to more clearly compare the influences of aftershock ground motions with different relative intensities (γ) on C_y, the ratio of C_y of the sequence-type ground motion to that of the corresponding mainshock (denoted as C_y,seq/C_y,ms) is calculated. Figure 9 illustrates the mean C_y,seq/C_y,ms for the system with μ_u = 6 and D = 1.0 under seismic sequences with different γ values. The C_y,seq/C_y,ms increases with the increase in γ. Moreover, the influence of aftershock on the strength demands of a structure is more significant in the short period. Consider γ = 1.5 as an example. The value of C_y,seq/C_y,ms reaches 1.45–1.50 in the short period region, decreases with increasing period, and is approximately 1.25–1.30 in the long period.

Fig. 9

C_y,seq/C_y,ms with different γ values, μ_u = 6, D = 1.0: a site class B, b site class C

When γ is 0.5, the difference between the strength demand of the structure under a seismic sequence and that of the structure under the corresponding mainshock is less than 10%. Under this condition, the influence of aftershock on the strength demand can thus be neglected. When γ is 1.0, the difference between the two is 10–20%, the aftershock has a considerable influence on the strength demand, and the structure that is designed without considering the influence of aftershocks is unsafe.

6.2 Effect of cumulative damage

To investigate the effect of cumulative damage on the YPS, the displacement ductility and modified Park–Ang model are used as indicators to measure the damage degree of structures. The ductility-based C_y,μ and damage-based C_y,D under the same seismic sequence are calculated, as shown in Fig. 10. When D and μ_u are the same, the trend of C_y,D and C_y,μ with the change in period remains basically the same. Because of the contribution of energy to structural damage, C_y,μ is always less than C_y,D.

Fig. 10

Comparison between C_y,D (D = 1.0) and C_y,μ: a site class B, b Site class C

To quantitatively reflect the difference between C_y,D and C_y,μ, C_y,μ/C_y,D with different ductility factors under seismic sequence and γ = 0.5 is studied, as shown in Fig. 11. It is indicated that the value of C_y,μ/C_y,D is 0.7–0.9 for the low ductility (μ_u = 2) and 0.6–0.9 for the high ductility (μ_u = 6) under sequence-type ground motions.

Fig. 11

C_y,μ/C_y,D with γ = 0.5: a site class B, b site class C

In short period region, the value of C_y,μ/C_y,D under sequence-type ground motions with γ = 0.5 sharply decreases with increasing period. This indicates that the proportion of the hysteresis phase in the modified Park–Ang model drastically changes. The hysteretic energy dissipation is relatively small because the structural yield strength tends to be the same as the strength demand of the elastic structure when the period approaches zero. The difference between C_y,μ and C_y,D is therefore small.

In medium and long period regions, C_y,μ/C_y,D under sequence-type ground motions with γ = 0.5 slightly changes with increasing period. The reason is that the proportion of the energy phase in damage index slightly increases with increasing period. With the increase in μ_u, the hysteretic energy dissipation of the structure under seismic sequences increases, and the difference between C_y,μ and C_y,D is more considerable.

6.3 Effect of site categories

The effect of site categories on the C_y can be observed in Figs. 3, 4, 5, 6, where C_y spectra are plotted under seismic sequences, which are recorded on site classes B and C. It can be seen that the spectra of C_y on the two site classes have similar tendencies in whole period ranges.

For comparison, the ratio of the yield strength coefficient (C_y,B) for site class B to that of C_y,C for site class C is calculated and plotted in Fig. 12. It can be observed that the difference between C_y,B and C_y,C can reach 20%. Site class B exhibits higher C_y values in periods 0–1.0 s and 4.5–6.0 s, whereas it exhibits lower C_y values in the period 1.0–4.5 s. This means that when the local site effect is neglected, a certain overestimation of the yield strength demand (C_y,B) results in periods 0–1.0 s and 4.5–6.0. A trend opposite the foregoing, however, is exhibited by C_y,C. It is thus evident, that impact of site conditions should be considered in seismic design. It is further observed that C_y,B/C_y,C is relatively independent from ductility and damage index.

Fig. 12

Effect of site class on YPS with γ = 0.5: aC_y,B/C_y,C, D = 1.0, bC_y,B/C_y,C, μ_u = 6

6.4 Effect of damping

To study the damping effect, the C_y values with damping ratios of ζ = 0.02 and ζ = 0.10 are calculated. The C_y values of these systems are normalized by the C_y values with damping ratio 0.05. Figure 13 presents the mean normalized C_y values of sequence-type ground motions with γ = 0.5 for each μ_u and D.

Fig. 13

C_y,ζ/C_y,ζ=0.05 with γ = 0.5: aD = 1.0, site class B, bμ_u = 6, site class B, cD = 1.0, site class C, dμ_u = 6, site class C

It can be observed that the increase in damping ratio always results in the decrease in C_y by various extents. For elastic structures, the input energy of earthquake is dissipated mainly through damping. And for inelastic structures, in addition to damping, hysteretic energy is also an important factor to dissipate the input energy of earthquake. As damping increases, the damping energy also increases, and the displacement response and hysteretic energy of structures subjected to the same ground motions decrease. Damping has a considerable effect on elastic structures but a minimal effect on inelastic structures.

Consider the C_y value of a structure with ζ = 0.05 as benchmark. The influence levels of damping are typically 30% and 23% when ζ = 0.02 and 0.10, respectively. With the decrease in ductility factor and damage index, the corresponding C_y when ζ = 0.02 or 0.10 approaches the C_y value when ζ = 0.05.

6.5 Effect of post-yield stiffness

In order to investigate the influence of post-yield stiffness ratio (H) on C_y, the C_y values of systems with H = 0.05 and 0.1 are calculated. Thereafter, these C_y values are normalized by the EPP system C_y for each sequence-type ground motion. Finally, the mean normalized C_y of all sequence-type ground motions with a constant γ value is calculated for each period.

Figure 14 shows the mean normalized C_y of sequence-type ground motions with γ = 0.5. It can be observed that the EPP system C_y is 1.0–1.1 times the C_y value of the system with H = 0.05. The C_y value of the latter is 1.0–1.05 times the C_y value of the system with H = 0.10. The results indicate that although an increase in this ratio leads to a slight increase in C_y, it is not the major influencing factor.

Fig. 14

YPS with different post-yield stiffness ratios with γ = 0.5: aC_y,H/C_y,H=0, site class B, bC_y,H/C_y,H=0, site class C

7 Predictive model

The predictive model of the YPS is an effective tool in seismic design for determining the capacity demand of structures subjected to seismic sequences. According to the statistical results, the factors that affect the YPS are period, ductility factor, damage index, aftershock intensity, site, and damping ratio. These factors are therefore also included in the predictive model of YPS:

$$C_{ye} = \left\{ {\begin{array}{*{20}l} {\left( {b_{0} + b_{1} T} \right)\eta_{\gamma } \eta_{\zeta } , \quad T \le 0.2s} \\ {{\text{linear }}{\text{int}} {\text{erpolation}}, \quad 0.2s < T \le 0.4s} \\ {\frac{1}{{b_{2} + b_{3} T^{1.1} }}\eta_{\gamma } \eta_{\zeta }, \quad 0.4s < T \le 6.0s} \\ \end{array} } \right.$$

(4)

$$R_{D} = 1 + \frac{{D(\mu - 1)(a_{0} T + a_{0} T^{2} )}}{{(\mu + a_{2} )(1 + a_{3} T + a_{4} T^{2} )}} \cdot \frac{1}{{0.87 + 0.08e^{r} }}$$

(5)

$$C_{y} { = }\frac{{C_{ye} }}{{R_{D} }}$$

(6)

$$x_{y} = \frac{{T^{2} C_{y} g}}{{4\pi^{2} }}$$

(7)

where T is the period; μ is the ductility factor; D is the damage index; C_ye is the pseudo-acceleration response spectra for sequence-type ground motions; C_y is the yield strength coefficient; R_D is the damage-based strength reduction factor, which can be obtained from the research of Sun C X et al. (2016) and Zhang et al. (2017); parameters a₀–a₄ are found in literature (Zhang et al. 2017); η_ζ is the damping ratio influence factor expressed as

$$\eta_{\zeta } = 1 + \frac{0.05 - \zeta }{{0.16 + 3.2\zeta }}$$

(8)

where ζ denotes damping; η_γ is the influence factor of the aftershock intensity expressed as

$$\eta_{\gamma } = 0.93 + 0.07\gamma^{2.5}$$

(9)

where γ is the relative intensity of aftershock.

The statistical data of YPS are used for the regression analysis. Parameters b₀, b₁, b₂, and b₃, which are calculated by a nonlinear least square regression analysis method, are regression parameters that depend on site classes. Table 3 summarizes the calculated values of these regression parameters.

Table 3 Values of b₀–b₃

The yield point spectra, which are predicted using Eq. (4) – (7), are compared with recorded mean spectra with γ = 1.0, as shown in Fig. 15. All ductility factors and damage indices are in good agreement.

Fig. 15

Comparison of calculated modified YPS with original spectra with γ = 1.0: aD = 1.0, site class B, bμ_u = 6, site class B, cD = 1.0, site class C, dμ_u = 6, site class C

8 Conclusions

The objective of this paper is to propose the damage-based yield point spectra (YPS) for mainshock-aftershock sequence-type ground motions. In the case of considering cumulative damage, the strength demands of inelastic systems can be determined more reasonably by damage-based YPS which determined for different damage indexes and ductility factors. A statistical study of the YPS is accordingly conducted. The yield point spectra are calculated for a series of elastic–plastic SDOF systems with various damage indices and ductility factors and subjected to 342 seismic sequences recorded under different site classes. In particular, the effect of aftershocks on the YPS is studied. The conclusions are as follows:

• 1. The yield point spectra in the short and medium period regions are strongly dependent on the period, whereas those in the long period are relatively independent from the period. In the entire period region, the yield point spectra decrease with increasing damage index (D) and ductility factor (μ_u).

• 2. There is a big difference between damage-based YPS and ductility-based YPS. The former is 40% higher than the latter in the long period with μ_u = 6 and γ = 0.5. The yield strength coefficient demand of C_y,D is greater than that of C_y,μ.

• 3. The influence of aftershock on the YPS increases with increasing aftershock intensity. The aftershock with γ = 0.5 has a negligible effect on YPS, while the aftershock with γ = 0.5 can increase the YPS to a 50% in the short period region. The effect of aftershock ground motion on the YPS depends on structural period, ductility factor, damage index, and aftershock intensity.

• 4. The predictive model of damage-based YPS is put forward, which is a function of period, ductility factor, damage index, damping, and aftershock intensity. The parameters in model are rely on site classes and hysteretic models. The predictive model can provide a good estimate of the damage-based YPS.