Major seismic events are commonly followed by a series of aftershocks, whose effects on slope stability should not be underestimated. Observations of earthquake-induced damage indicate that aftershocks often act as the final trigger for slope failure (Xu and Huang, 2008; Yin, 2008). As the safety requirements for engineering structures become increasingly stringent, slope stability under MAS has attracted considerable interest from both researchers and engineering practitioners. Li et al. (2009) developed and improved the PDEM framework for analyzing stochastic dynamic systems, providing a systematic approach to nonlinear stochastic dynamic analysis and reliability assessment of large-scale and complex engineering structures. The framework has been widely applied to stochastic dynamic analyses of slopes (Li and Chen, 2017). Building on this work, Pang et al. (2021, 2024) introduced an enhanced generalized-PDEM approach for reliability analysis of complex slopes. Their methodology considers multiple slope parameters and ground motion uncertainties. Additionally, accounting for the spatial variation in soil strength properties, a reliability assessment methodology integrating the Newmark method and PDEM is developed to evaluate the effects of aftershocks and spatial heterogeneity on slope dynamic reliability (Newmark, 1965; Zhou et al., 2023; Wang et al., 2022; Xu et al., 2025). Most existing studies use peak ground acceleration (PGA) as the primary measure of ground motion intensity. Nevertheless, intensity measures based only on peak values may not adequately capture the overall characteristics of MAS ground motions (Ruiz-Garcia and Negrete-Manriquez, 2011; Amiri et al., 2022).

For characterizing the ground motion random field of the MAS, the Monte Carlo method is a well-established and effective approach (Metropolis and Ulam, 1949; Hu et al., 2018; Nithin et al., 2020; Kim and Sitar, 2013). However, it requires a large number of samples and incomplete probability information of the sample set (Jiang et al., 2021). The random function-dimension reduction simulation technique generates MAS time histories with associated probabilities, thereby forming a complete probabilistic set. These time histories can then be coupled with the PDEM to conduct sophisticated dynamic response and reliability analyses for complex engineering structures subjected to MAS (Liu and Liu, 2017; Liu et al., 2019, 2022).

Current reliability analysis methods for seismic slopes are predominantly developed for single-mainshock scenarios and therefore may not adequately assess slope reliability under MAS. Moreover, existing studies on sequential ground motions predominantly use PGA as the intensity measure, which may not adequately capture the cumulative damage effect contributed by aftershocks. To address these limitations, this study adopts cumulative absolute velocity (CAV) as the intensity measure for MAS ground motions and develops a reliability analysis framework for the permanent displacement of soil slopes based on the PDEM. The proposed method accounts for the effects of aftershocks and provides a basis for stability assessment and disaster prevention of soil slopes subjected to MAS.

Empirical MAS recordings provide more realistic representations of structural damage and cumulative degradation (Yang et al., 2022). This study uses a diverse set of real earthquake records to reduce the bias associated with specific ground motion types. Through iterative parameter identification based on evolutionary power spectral density modeling, this study combines theoretical models with data-driven methods to construct stochastic seismic inputs. These inputs provide a stochastic loading basis for investigating the effects of MAS on slope reliability.

This study selected 96 recorded mainshock–aftershock pairs from 15 seismic events in the NGA-West2 strong-motion database maintained by the Pacific Earthquake Engineering Research Center (PEER). We selected records based on five criteria (Bray and Macedo, 2019; Yeznabad et al., 2022; Yeznabad et al., 2026): (1) the mainshock and its aftershock must belong to the same seismic event, with the aftershock defined as the largest-magnitude event occurring within 12 months after the mainshock; (2) both events have a Mw ≥ 5.0 and a PGA ≥ 0.05 g; (3) both records are taken from the same component of the same station; (4) the station has an average shear-wave velocity in the upper 30 m (Vs30) between 100 and 700 ms-1; and (5) the rupture distance of each event does not exceed 80 km. The 96 MAS records selected according to the above criteria encompass a broad range of sequence-type ground-motion characteristics. However, as these records are derived from the NGA-West2 database and primarily represent shallow crustal earthquakes, the applicability of the results is primarily limited to MAS ground motions from shallow crustal earthquakes.

Each MAS adopted a “mainshock + 20 s temporal gap + aftershock” configuration to preserve comprehensive non-stationary sequence characteristics (Wang et al., 2022). Figure 3 displays a typical mainshock–aftershock acceleration record from the Whittier Narrows event, with the mainshock and aftershock numbered 589 and 707, respectively, in the NGA-West2 database. In the sensitivity analysis of the 21 MAS ground-motion parameters, the original ground-motion records were used without amplitude scaling. Subsequently, when calculating the permanent displacement of the slope at different CAV levels, the ground-motion amplitudes were scaled to better represent practical engineering conditions.

For the mainshock–aftershock sequences constructed above, frequency-domain energy distribution analysis is performed using an evolutionary power spectrum model for fully non-stationary ground motion processes (Priestley, 1965; Liu and Liu, 2017). The best square approximation criterion and the least square method are adopted to fit the energy distribution curve in the frequency domain. The parameter vector λ is further obtained through inversion and regression. In the process, considering the site damping ratio ξf excellence, site soil circular frequency ωf and the influence of a frequency modulation function parameter. The identification results are shown in Fig. 4, and the corresponding data are listed in Table 2. The constructed ground motions can be regarded as zero-mean, real-valued, non-stationary ground motion processes generated by the evolutionary power spectral density function model.

To assess slope reliability based on permanent displacement responses, it is necessary to define critical displacement thresholds. Jibson and Michael (2009) classified slope displacement into four ranges for seismic landslide risk evaluation: displacements of 0–1 cm indicate low risk, 1–5 cm indicate medium risk, 5–15 cm indicate high risk, and those exceeding 15 cm indicate very high risk. For permanent displacement, Ozkan (1998) suggests that the control standard for seismic sliding deformation is 1 m. In this study, based on the actual engineering requirements, three slope displacement thresholds of 0.05, 0.25, and 0.50 m were selected, corresponding to low-, medium-, and high-level failure states, respectively, to systematically evaluate slope reliability under MAS.

Figure 5 shows the distribution of slope permanent displacement under MAS. The distribution is widely dispersed, and the slope displacement response increases with increasing ground motion intensity. When the CAV of the MAS was 12, 28 and 40 ms-1, respectively, the corresponding average slope permanent displacements were 0.067, 0.328 and 0.633 m, respectively, which are comparable to the slope displacement results when the PGA of the MAS was 0.4, 0.5 and 0.6 g in reference (Wang et al., 2022). Figure 6 compares the upper bounds of the permanent displacement ranges obtained at three corresponding CAV and PGA levels. It can be observed that, compared with PGA, the use of CAV leads to a more concentrated distribution of permanent displacement results, with the overall dispersion reduced by approximately 40 % on average. These results demonstrate that CAV provides a more concentrated representation of permanent displacement responses. Therefore, CAV is more suitable than the commonly used PGA for characterizing MAS ground motions in the reliability analysis of slope permanent displacement.

Nevertheless, even with the use of CAV as the MAS ground-motion intensity measure, the obtained slope displacement responses remain somewhat scattered. This suggests that additional ground-motion parameters – including frequency content and duration – also influence slope permanent displacement (Rathje and Saygili, 2008, 2009; Yeznabad et al.,2022). Given the current lack of a single comprehensive ground-motion parameter that can effectively predict slope displacement responses, the PDEM framework propagates this inherent variability into the reliability estimate, thus rendering a probabilistic treatment indispensable. Building on previous studies, this study develops a reliability probabilistic analysis method based on the PDEM.

Figure 7 presents the PDF and CDF curves of slope permanent displacement. The PDF curves show the distribution characteristics and main concentration ranges of slope permanent displacement, while the CDF curves provide the cumulative probability at different displacement levels for evaluating slope reliability based on the specified displacement thresholds. The PDF curves shown in Fig. 7a exhibit bimodal or multimodal characteristics, making them difficult to describe using standard distributions such as the normal or lognormal distribution. Under the MAS action of different intensities, the displacements of slopes have diverse distribution patterns, and their probabilities are also not the same. When the CAV is 12 ms-1, slope permanent displacement ranges from 0 to 0.25 m, and the slope reliability is 48.5 % at the displacement threshold of 0.05 m corresponding to the low failure state. At a CAV of 28 ms-1, the displacement range expands to 0–1.00 m, and the slope reliability is 45.5 % at the displacement threshold of 0.25 m corresponding to the medium failure state. At a CAV of 40 ms-1, the displacement range further expands to 0–2.00 m, and the slope reliability decreases to 31.7 % at the displacement threshold of 0.50 m corresponding to the high failure state.

Figures 8 and 9 compares the CDF of slope permanent displacement obtained using CAV and PGA. The CAV-based CDF curves show an overall leftward shift relative to the corresponding PGA-based curves, indicating smaller displacement values at the same cumulative probability. At a cumulative probability of 50 %, the slope permanent displacements are approximately 0.07, 0.32, and 0.55 m for CAV = 12, 28, and 40 ms-1, respectively, compared with approximately 0.20, 0.50, and 0.70 m for PGA = 0.4, 0.5, and 0.6 g. The difference becomes more evident at higher cumulative probabilities. The PGA-based CDF exhibit more pronounced large-displacement tails and greater dispersion, whereas the CAV-based CDF show more concentrated displacement distributions. Overall, CAV better captures the cumulative effects of MAS on slope permanent displacement and therefore provides a more comprehensive intensity measure for slope reliability analysis under MAS.

By identifying the intensity measure of MAS and integrating it with the PDEM, this study develops a reliability analysis method for slope permanent displacement under MAS. The main conclusions are as follows: 1.

Among the 21 MAS ground-motion parameters considered, CAV showed the strongest correlation with slope permanent displacement (ρ = 0.872) and best represents the cumulative effects of MAS. Therefore, CAV can be used as the intensity measure to assess the response of soil slopes under MAS.

This study focuses on clayey soil slopes in southwestern China and examines the feasibility of the proposed reliability analysis method for soil slope responses under shallow-crustal MAS. The effects of soil-parameter variability and slope geometric variability are not considered. The principal contribution of this study lies in the proposed methodological framework, which integrates CAV as the intensity measure for MAS with the PDEM for efficient reliability analysis. Further consideration of soil-parameter variability is expected to improve the applicability of the proposed method to practical engineering problems.