9. Conclusion and Prospects

Building upon a summary of commonly employed probability analysis methods, this study introduces the Generalized Probability Density Evolution (GPDE) method into the realm of random dynamics and probability analysis for high-faced rockfill dams. Firstly, it is emphasized that under seismic actions, high-faced rockfill dams exhibit numerous sources of uncertainty, primarily stemming from the stochastic nature of seismic motion and the uncertainty in material parameters. Notably, there is a scarcity of research that analyzes the seismic response of high-faced rockfill dams from the perspectives of random dynamic time history and probability. Subsequently, a critical review of prevalent methodologies such as the first- and second-order moment method, Monte Carlo method, and response surface method is undertaken, highlighting their limitations in probabilistic seismic analysis for complex geotechnical engineering, including high-faced rockfill dams. To address these shortcomings, the study proposes the Generalized Probability Density Evolution method based on stochastic dynamics theory, offering applicability to the probabilistic seismic response and analysis of complex geotechnical engineering structures, including high-faced rockfill dams. Finally, the application and solution processes of the Generalized Probability Density Evolution method are elaborated upon. This includes the establishment of a fully non-stationary random seismic motion model based on spectral representation-random function and a high-dimensional random sample parameter generation method employing the GF-deviation optimization point selection technique. Through analytical solutions, Duffing oscillator, multi-layered rock-soil slopes, and equivalent linear random dynamic and probabilistic analyses of structures such as high-faced rockfill dams, the effectiveness and reliability of the combined GPDE method with stochastic seismic motion and high-dimensional random parameter generation methods are validated. This contributes to laying the foundation for the random dynamic analysis of high-faced rockfill dams and performance-based seismic safety assessment in complex geotechnical engineering.

Taking into full consideration the stochastic nature of seismic motion, this study unveils the random dynamic characteristics of high-faced rockfill dams and establishes a performance-based seismic safety assessment process. Initially, a comprehensive set of seismic acceleration time histories with complete probability information is discretely generated. A series of elastic–plastic finite element dynamic analyses is conducted, and in conjunction with the Generalized Probability Density Evolution theory, the random seismic response and extensive probability information of high-faced rockfill dams are obtained. Subsequently, from the perspectives of random dynamics and probability, the study elucidates the variations and distribution characteristics of dam body acceleration, deformation, and panel stress. It indicates that different seismic intensities and seismic actions have a significant impact on the seismic response of high-faced rockfill dams. For a given set of seismic actions, the maximum response can be three to five times the minimum response. Therefore, a comprehensive assessment of seismic safety for high-faced rockfill dams is essential from the viewpoint of seismic motion randomness. The study also reveals the spatially irregular flow characteristics of the probability responses and the non-normal distribution features, such as non-Gaussian distributions. Based on the 5%, 50%, and 95% exceedance probabilities, the study obtains the ranges of values for various physical quantities, providing reference values for numerical calculations and the analysis of the ultimate seismic capacity of high-faced rockfill dams. Finally, performance level division standards are proposed based on dam body deformation and panel impermeable body safety. These standards are validated for their reasonability through probability assurance. Specifically, for dam body deformation, the performance indicators of 0.3%, 0.7, and 1.0% relative settlement at the dam crest correspond to slight damage, moderate damage, and severe damage threshold states, aligning well with the damage observed in the Zipingpu face rockfill dam during the Wenchuan earthquake. A standard of 1.1% is suggested as the non-failure criterion. For panel impermeable body safety, a dual-control performance indicator is proposed, combining the demand stress ratio with the cumulative over-stress duration. The study establishes a multi-seismic intensity-multi-performance target-exceedance probability performance relationship, constructs vulnerability curves, and outlines a preliminary performance-based seismic safety assessment method for high-faced rockfill dams.

A high-dimensional elastoplastic random parameter sample generation method is established to reveal the stochastic dynamic response and probabilistic characteristics of high-faced rockfill dams under the influence of random material parameters. Initially, elastoplastic random parameter samples are generated based on the GF-deviation optimization point selection method. In conjunction with the Generalized Probability Density Evolution method, the study, from both the perspectives of random dynamics and probability, asserts that under deterministic seismic actions (PGA = 0.5 g), the uncertainty in material parameters significantly impacts the seismic response of high-faced rockfill dams, affecting dam body acceleration, deformation, and panel stress. Subsequently, using the 5% exceedance probability and 95% exceedance probability as benchmarks, it is noted that the maximum response is approximately 1.5–2 times the minimum response, indicating a relatively minor impact on seismic response variability compared to the randomness induced by seismic motion. Lastly, through a comparison of response values considering material parameters with normal distribution and log-normal distribution, the study indicates that the distribution type has a negligible effect on seismic response, with differences generally within 10%. Consequently, under deterministic seismic excitation, considering the stochastic nature of material parameters is essential, but the influence of distribution type needs to be considered judiciously.

The seismic motion-material parameter coupling randomness is systematically considered, and its impact on the dynamic response and seismic safety of high-faced rockfill dams is comprehensively studied from the perspectives of random dynamics and probability. This enhances the performance-based seismic safety evaluation framework. Initially, utilizing spectral representation-random function and random material parameter variables, random seismic acceleration time histories and random material parameter samples are concurrently generated. Subsequently, a detailed investigation from the viewpoints of random dynamics and probability is conducted on the seismic response of high-faced rockfill dams under the influence of different seismic intensities, considering the coupling randomness of seismic motion and material parameters. Through a comparative analysis of the effects of seismic motion randomness, material parameter randomness, and coupled randomness on dam body acceleration, deformation, and panel stress, it is evident that seismic motion primarily controls seismic response. Therefore, in a comprehensive analysis of the random dynamic response of dams, the influence of material parameter randomness can be to some extent neglected. Finally, a performance relationship is established under coupled random factors, encompassing multiple seismic intensities, multiple performance objectives, and exceedance probabilities. Vulnerability curves for different damage levels are obtained, indicating that the differences in performance safety assessment probabilities obtained from seismic motion randomness and coupled randomness are minimal. In the context of performance-based seismic safety assessment for high-faced rockfill dams, consideration of seismic motion randomness alone is deemed sufficient to meet the required criteria.

From the perspective of three-dimensional elastoplastic analysis, this study investigates the performance indicators and performance levels for panel failure evaluation based on the ratio of accumulated overstress volume combined with the cumulative overstress time. This contributes to the further refinement of the performance-based seismic safety evaluation framework. Initially, building upon the previous research findings and considering the stochastic nature of seismic motion, the study, from the standpoint of random dynamics, elucidates the three-dimensional effects on dam body acceleration, deformation, and panel stress. This underscores the necessity of considering seismic motion randomness in seismic response analysis and highlights the significance of response distribution patterns and ranges for the seismic safety assessment and ultimate seismic capacity analysis of high-faced rockfill dams. Subsequently, the study explores and suggests preliminary performance indicators and performance levels for panel seismic safety evaluation based on the ratio of accumulated overstress volume combined with cumulative overstress time. The rationality of performance level division is demonstrated through probability assurance. Finally, the study reveals the probability relationship between two-dimensional deformation and three-dimensional deformation, obtaining vulnerability curves for each performance indicator. This further enhances the performance-based seismic safety evaluation framework, providing a scientific basis for the seismic design and performance control of high-faced rockfill dams.

This study systematically considers the randomness of seismic motion, the uncertainty of material parameters, the coupling randomness of seismic motion and material parameters, and the combined effect of rockfill softening. It explores the performance-based seismic safety evaluation framework for the seismic stability of high-faced rockfill dam slopes from the perspectives of random dynamics and probability. Firstly, through stochastic dynamic and probability analyses, the study reveals that seismic events, especially strong earthquakes, significantly impact the safety factor, cumulative time exceeding the safety factor, and cumulative slip of dam slopes due to the rockfill softening effect. As seismic intensity and duration increase, the softening characteristics gradually become more pronounced, presenting a progressive process. Reliability analysis indicates that evaluating the stability of soil-rock dam slopes solely based on the minimum safety factor is unreasonable. Instead, a comprehensive assessment of slope seismic safety is necessary, considering cumulative time exceeding the safety factor and cumulative slip. This approach holds crucial significance for performance-based seismic safety evaluation of high-faced rockfill dams. Subsequently, the study uncovers the influence and relationships of seismic motion randomness, material parameter uncertainty, and seismic motion-material parameter coupling randomness on the safety factor, cumulative time exceeding the safety factor, and cumulative slip of dam slopes. The relationship between cumulative time and cumulative slip is also discussed. The necessity of analyzing dynamic slope stability from the perspective of random dynamics and the importance of performance-based seismic safety evaluation are emphasized. This provides valuable insights for dam slope anti-sliding design and the analysis of ultimate seismic capacity. Finally, performance level division standards for two performance indicators, namely cumulative time exceeding the safety factor and cumulative slip, are proposed as follows: cumulative time of 0 s (sliding threshold), 0.5, and 1.5 s; cumulative slip of 0 cm (minimal sliding), 20, and 100 cm. These correspond to the threshold states of slight damage, moderate damage, and severe damage, respectively. The study establishes a multi-seismic intensity-multi-performance target-exceedance probability performance relationship and vulnerability curves. These serve as references for dam slope stability safety assessment, contributing to the further refinement of the performance-based seismic safety evaluation framework for high-faced rockfill dams.

In-depth exploration of seismic motion uncertainties and refined identification of strong earthquakes. Seismic motion is a complex load with significant uncertainties. Regardless of seismic motion frequency, peak values, amplitude variations, duration, and the arrangement sequence of different impulses, it is essential to further investigate the uncertainties in seismic motion characteristics. This includes probabilistic insights into its impact on the dynamic response of high-faced rockfill dams, encompassing seismic hazard analyses. Additionally, the development of simple and rational seismic motion parameter indicators is a key focus. Identifying strong earthquakes, such as impulse and sequential earthquakes, which have a substantial impact on high-faced rockfill dams, and exploring their uncertainties in dynamic effects, is another research direction.

Statistical analysis and refined uncertainty analysis of pile material model parameters. Based on relevant experiments and engineering cases, enhancing the statistical characterization of pile material model parameters and establishing their distribution models, exploring the correlation between parameters, and developing methods considering spatial variability of pile material are essential. This aims to provide a more detailed understanding of the impact of pile material parameter randomness and the coupling randomness of seismic motion-material parameters on the dynamic response of high-faced rockfill dams.

Research on fast and refined numerical analysis methods. Nonlinear dynamic time history analysis involves significant computational complexity. Therefore, ongoing efforts are required to develop efficient three-dimensional dynamic nonlinear time history analysis methods for the coupled system of reservoir water, dam, and foundation. This includes detailed analysis, such as the elastic–plastic damage analysis of panel failure, which is of great significance for achieving performance-based seismic safety assessment.

Improvement of performance-based seismic safety assessment methods. Currently, for dam deformation, impermeable body safety, and dam slope stability, general methods are based on the numerical constitutive framework of elastic–plastic or equivalent linear models. There is a need to establish a unified numerical simulation analysis framework. Furthermore, proposing comprehensive performance evaluation indicators and defining performance levels are crucial steps toward establishing and refining the performance-based seismic safety assessment system for high-faced rockfill dams.