Reinforced concrete (RC) frame structures, as a structural system that enables flexible space separation, are widely used in Chinese cities. Previous earthquake damage surveys have shown that reinforced concrete frame structures are often severely damaged or even collapse during earthquakes [1–4]. Therefore, accurate and solid seismic fragility analysis of RC frame structures is important and necessary. Seismic fragility analysis is an effective method for evaluating the seismic performance of a structure from a probabilistic perspective [5–10].

The first and important step in the fragility analysis is to select a ground motion intensity measure (IM) to measure the aleatory randomness caused by record-to-record variability. The seismic IM is used as an intermediate variable to connect the seismic hazard and engineering performance demand measure (DM). To date, a variety of ground motion IMs have been proposed, and the selection of IMs depends not only on the structural system concerned but also on the engineering performance DM of interest [11–15]. Existing studies are mainly focused on the macro analysis of the relationship between ground motion IMs and engineering performance DMs. At the same time, the applicability of different IMs for seismic fragility analysis at various damage limit states has not yet been analyzed in depth.

The seismic fragility curve illustrates a structure's probability of exceeding a particular damage state under the action of earthquakes with different intensities. However, the establishment of fragility curves requires a significant amount of computations. Therefore, it is critical to quickly and easily estimate fragility curves. Commonly used methods for establishing fragility curves are incremental dynamic analysis [16–20], truncated incremental dynamic analysis [21], multiple stripes analysis [22,23], and static pushover analysis methods [24–26]. Compared with incremental dynamic analysis, the multiple stripes analysis method does not need to scale the amplitude of all the ground motions to a level that causes the damage limit state of interest, but only needs to analyze the structure at specific ground motion intensity levels [21]. The multiple stripes analysis method is equivalent to a special case of the IDA method. Even compared with the truncated incremental dynamic analysis, the calculation amount of the multiple stripes analysis is still smaller [21]. The multiple stripes analysis method can consider the influence of high-modes characteristics of high-rise structures and the hysteretic characteristics of structural members compared with static pushover analysis methods. Therefore, the multiple stripes analysis method can easily present the fragility curves while ensuring accuracy compared to other methods.

Spectral acceleration at fundamental period S_a(T₁), a convenient and efficient intensity measure for first mode dominated structures [27], has been widely adopted in seismic design codes and research in many countries. In recent years, various forms of IMs have been proposed based on S_a(T₁). The purpose of this paper is to investigate the estimation of aleatory randomness by using different S_a(T₁)-based ground motion IMs in the fragility analysis in terms of different damage limit states for the RC frame structures. With thirty pairs of far-field records, multiple stripes analysis is carried out for two RC frame structures with 4 and 8 stories to explore the effect of S_a(T₁)-based IMs on the record-to-record variability estimated in fragility curves of different damage limit states. The influence of the bidirectional ground motion intensity characterization methods on the seismic fragility assessment is also explored. Furthermore, the optimal parameters in the dual-parameter ground motion IM for the fragility analysis at different damage states are proposed through parameter analysis. Finally, the improved dual-parameter ground motion IM is applied to the risk assessment of exceeding each limit state for the 8-story frame structure according to the Chinses seismic code.

There are many types of ground motion IMs, and the application scopes of different ground motion IMs are different [11,12]. Due to the structural characteristics of RC frame structures, frame buildings in seismic high-risk areas are often structurally not high, and then their dynamic characteristics are generally dominated by the first vibration mode. The spectral acceleration at the first vibration period S_a(T₁) is a convenient and efficient ground motion IM in the seismic analysis of structures dominated by the first mode [27]. However, RC frame structures are prone to be elasto-plastic under the action of strong earthquakes, and a single S_a(T₁) cannot reflect this characteristic. Cordova et al. [28] considered the effect of structural stiffness degradation and proposed the dual-parameter ground motion IM S* (see Table 1), where the softened period T_f = 2T₁ and the combination coefficient α = 0.5. Mehanny [29] proposed the improved IM-CR and IM-SR based on S*. By introducing the self-adaptive parameter R_IM, both IM-CR and IM-SR can be applied to situations with different nonlinear levels. R_IM was recommended to be 2. Bojórquez et al. [30] proposed an IM exploring the geometric mean of spectral acceleration at multiple periods, in which the parameter N_p is used to capture the spectral shape, and T_N = 2T₁ and α = 0.4 are recommended.

However, the abovementioned ground motion IMs only capture the effect of period elongation and do not reflect the effect of higher vibration modes for long-period structures. Some studies have pointed out that ground motion IMs considering the higher vibration mode are suitable for high-rise buildings. The linear combination-type IMs Sa12∗ and Sa123∗ proposed by Shome et al. [27] considers the spectral accelerations at the first two and three periods, respectively, and the combination coefficients are the weighted average of the modal mass participation coefficients of a 20-story building. Vamvatsikos et al. [31] proposed IMs in flexible combination forms, namely, IM₁₂ and IM₁₂₃. For IM₁₂, the power exponent β is set as 1/2. The parameters τ_a and τ_b are suggested to be T₁ and T_f, respectively, when IM₁₂ is used in low-rise buildings, that is, IM₁₂ is equivalent to S* in such cases. The parameters τ_a and τ_b are suggested to be T₁ and T₂, respectively, when IM₁₂ is used in medium-rise buildings. For IM₁₂₃, the power exponents β and γ are recommended to be 1/3, and the first three periods T₁, T₂ and T₃ are included in terms of high-rise buildings. Similarly, Lin et al. [32] proposed IM S_N1 considering the softened period and IM S_N2 considering the first two vibration periods T₁ and T₂, and suggested C = 1.5, α = 0.5 and β = 0.75. Lu et al. [33] investigated the relationship between the optimal number of combined modes n and the first vibration period T₁ and proposed ground motion IM S¯a. Kazantzi et al. [34] proposed ground motion IM S_a,gm(T_i) based on the geometric mean form considering spectral acceleration at multiple periods for building classes. It is pointed out that the IM including five spectral values, (T_i)₅, can be well applied to low and high building classes. To assess the collapse capacity of generic moment frames, Adam et al. [35] proposed an optimal IM_opt that includes an unfixed lower bound period T_0.95M. T_0.95M can be estimated as T0.95M≈T1/[1+3(m0.95M−1)/2], in which m0.95M=ceilN is the mode associated with the exceedance of 95% of the total mass and N is the number of stories in the building.

In addition to the fourteen IMs listed in Table 1, three simple amplitude-type IMs PGA, peak ground velocity (PGV) and peak ground displacement (PGD) are also evaluated here. For S*, α is set as 0.5 and T_f is set to be 2T₁. For IM-CR and IM-SR, α is set as 0.5 and R_IM is taken as 2. In I_Np, T_N is taken as 2T₁ and α is set as 0.4. The original parameters in Table 1 are used in Sa12∗ and Sa123∗. The first two vibration periods and the first three vibration periods are applied to IM₁₂ and IM₁₂₃, respectively. For S_N1, C is taken as 1.5 and α is taken as 0.5, and for S_N2, β is set to be 0.75. In S¯a, n is taken as 2. Five spectral values, (T_i)₅ = {T_2m, min[(T_2m + T_1m)/2,1.5 T_2m], T_1m, 1.5T_1m, 2T_1m}, is used in S_a,gm(T_i), in which the class-average vibrations periods, T_1m and T_2m, are taken as the first two periods. In IM_opt, the un-fixed lower bound period T_0.95M is estimated according to the aforementioned formula about the story number of the buildings.

Lucchini et al. [49] and Kostinakis et al. [50] pointed out that the characterization methods of bidirectional ground motion intensity using a single value influence the correlation between the ground motion IMs and engineering performance DM. Therefore, the influence of different characterization methods on the record-to-record variability in the fragility analysis is also investigated herein. As shown in Table 5, six characterization methods for bidirectional ground motion intensity are used here [50].

It should be noted that each ground motion has a different IM level when using different characterization methods or IMs other than PGA₁. In such a situation, to estimate the fragility curves, n_j in Eq. (1) is set to 1, and the probability of a particular limit state is set to 1 if θ_max is larger than the corresponding criterion. Taking S_a(T₁) as an example, the analysis processes of the 4-story frame using the two bidirectional characterization methods, i.e., IM₁ and IM_SRSS, are shown in Figs. 6 and 7, respectively. Figs. 6 and 7 show that β_RTR estimated by S_a(T₁) increases roughly with the shift of the limit state from IO to CP in both the bidirectional ground motion intensity characterization methods IM₁ and IM_SRSS.

Figs. 8 to 9 illustrate the uncertainties of the ground motion β_RTR estimated in three limit states of the two buildings using different bidirectional characterization methods. It can be seen from the figures that the limit state of the structure has an important influence on the applicability of the ground motion IM. With the development of the damage state, that is, from the IO limit state to the CP limit state, the uncertainties of the ground motion β_RTR corresponding to each IM generally show an increasing trend.

For both 4- and 8-story buildings, the IMs considering the effect of higher modes, such as Sa12∗, Sa123∗, IM₁₂, IM₁₂₃, S_N2 and S¯a, are not advantageous with larger β_RTR compared to S_a(T₁) in the three limit states for most bidirectional characterization methods. This may be because the spectral acceleration corresponding to the higher modes is not highly correlated with the structural damage in low and medium buildings [43]. However, the IMs considering the effect of the softened period, such as S*, IM-CR, IM-SR, I_NP and S_N1, even the IM S_a,gm(T_i) which considers both the effect of the softened period and higher modes, have lower β_RTR relative to S_a(T₁) and those IMs considering higher modes in most cases. Moreover, as damage develops, the advantages of IMs considering the effect of the softened period become increasingly obvious, especially for S*. This may be because the effect of the softened period becomes increasingly significant with the deepening of nonlinearity and damage [51], and the correlation between the structural response and the spectral acceleration corresponding to the softened period is strengthening.

With respect to most bidirectional characterization methods, both dual-parameter IMs S* and S_N1 can maintain lower β_RTR in the three limit states for the two buildings. Although IM-CR, IM-SR and I_NP are also combination-type ground motion IMs that consider the effect of period elongation, their differences in the selected softened period or the power exponents from S* and S_N1 caused significant differences in the evaluation of ground motion uncertainty. It is worth noting that in the IO state, the β_RTR of S_N1, IM-CR and IM-SR is smaller than S*. The selected softened periods in S_N1, IM-CR and IM-SR are smaller than that of S*. This shows that the selected softened period should not be a fixed value in different damage states. As analyzed, when the appropriate softened period is selected, that is, the appropriate parameter C in T_f = CT₁, the smallest β_RTR can be obtained. For the three amplitude-type IMs, PGV has apparent advantages in the three limit states of the buildings. Although IM_opt considered the impact of the number of floors in the structure, its performance is not satisfactory, which may be due to the failure to consider the fundamental period of the structure.

The uncertainties estimated by a specific IM using different bidirectional ground motion intensity characterization methods are different, as shown in Figs. 8 and 9. It can be seen from the above analysis that S* and S_N1 have apparent advantages in estimating of the uncertainty of ground motion. Taking S* and S_N1 as examples, the β_RTR estimated in the six bidirectional characterization methods are illustrated in Figs. 10 and 11, respectively. Since only one direction of ground motion intensity is considered, there is a large difference between IM₁ and IM₂. Structural damage may be related to the intensity of ground motion in both directions. When considering the combination of the intensity in the two directions, IM_AMV, IM_GMV and IM_SRSS can predict relatively stable and similar β_RTR in the two structures for each limit state. The β_RTR obtained by the two methods is close to the minimum in most cases.

In this section, the improved IM is further applied to the risk assessment at the three limit states for the 8-story frame structure according to Chinese codes. Generally, the fragility curve corresponding to one limit state is assumed to be a lognormal cumulative distribution function (see Eq. (4)) as used above in Eq. (1). (4)P(Limitstate∣IM)=Φ(ln⁡(IM/η)βRTR) where P(Limit state∣IM) is the probability of exceeding one specific limit state caused by a ground motion at the intensity level IM, and η and β_RTR are the median capacity and lognormal standard deviation, respectively. The mean hazard curve, λ¯IM, as shown in Eq. (5), can be estimated by fitting multiple hazard data. (5)λ¯IM(IM)=k0IM−k where k₀ and k are two empirical factors. These S_a(T₁) and S_a(T_f) corresponding to the frequent earthquake (63.2% probability of exceedance in 50 years), the design basis earthquake (10% probability of exceedance in 50 years) and the maximum considered earthquake (2% probability of exceedance in 50 years) are obtained referring to the Code for Seismic Design of Buildings [38] for the regions with seismic precautionary intensity of 8. Then, the SIM∗ corresponding to the three limit states can be obtained by adopting the suggested parameters in Eq. (3). The annual probability of exceedance\; λ¯IM for each SIM∗ can be obtained by assuming that the ground motion intensity level is Poisson distributed. Subsequently, the seismic hazard curves with respect to the three limit states of the 8-story frame can be obtained by fitting Eq. (5) with three discrete points (SIM∗, λ¯IM) at each limit state, as illustrated in Fig. 16. Due to the different IMs used in each limit state, the hazard risk curve in each limit state is slightly different.

The annual probability of exceeding one specific limit state, as listed in Eq. (6), can be obtained through the total probability theory by combining the fragility curve (Eq. (4)) with the seismic hazard curve (Eq. (5)). (6)λlimitstate=∫P(Limitstate∣IM)⋅|dλ¯IM(IM)| Furthermore, Eq. (7) can be obtained by substituting and simplifying Eq. (6) [52]. It should be noted that there is no consideration of epistemic uncertainty in Eq. (7). (7)λlimitstate=λ¯IM(η)exp⁡(12k2βRTR2)=k0η−kexp⁡(12k2βRTR2)

Table 7 shows the annual probability of exceeding each limit state for the 8-story frame structure. Note that the median capacity η and lognormal standard deviation β_RTR are estimated by SIM∗ using the bidirectional intensity characterization method of IM_GMV, as shown in Fig. 17. For the 8-story frame structure, the annual probability of exceeding the CP limit state is 7.40 × 10⁻⁵, which is lower than the acceptable annual collapse probability of 1 × 10⁻⁴ for Category-C buildings determined in Zhang et al. [53].

Based on two low- and medium-rise RC frame structures, an investigation is first provided in this study regarding the estimation of record-to-record variability β_RTR by using S_a(T₁)-based IMs in the fragility curves of three limit states (i.e., IO, LS and CP). Subsequently, the optimal multiplier C and exponent α in the dual-parameter IM for different limit states are suggested through parameter analysis. Furthermore, the improved dual-parameter IM is applied to the risk assessment at the three limit states. Several observations can be reached from the case study on the RC frame structure, as follows: (1)

The limit state of the structure has an important influence on the applicability of S_a(T₁)-based IMs. From the IO limit state to the CP limit state, the uncertainties of the ground motion β_RTR corresponding to most IMs generally show an increasing trend. For low- and medium–rise RC frame structures, the S_a(T₁)-based IMs considering the effect of a softened period, i.e., S* and S_N1, can maintain a lower β_RTR in the three limit states compared to the S_a(T₁)-based IMs considering the effect of higher modes.

Only two typical low- and medium-rise frame buildings with 4 and 8 stories are employed here. In addition, only 30 pairs of far-field records are selected in this paper. More structure cases and ground motion records with different spectral characteristics, such as near-field records, are needed to verify the suggested parameters in the improved IM.