Ground seismic response analysis based on the probability density evolution method

Dynamic reliability analysis of slopes based on the probability density evolution method

Random function based spectral representation of stationary and non-stationary stochastic processes

Base isolation is one of the most efficient methods for seismic mitigation of building and bridge structures in high‐intensity areas. However, the base‐isolated structure designed according to current codes based on the local seismic hazard and a few station records might not guarantee the performance of the base isolation system and ensure the safety of the structure. Carrying out a probabilistic analysis of seismic mitigation of base‐isolated structures, considering the randomness associated with earthquake ground motions, is thus necessary. To this end, the stochastic seismic response and reliability analysis of base‐isolated structures subjected to stochastic ground motions are conducted in this study, in conjunction with the probability density evolution method and the principle of equivalent extreme value event. The recently developed sliding hydromagnetic bearing is employed to constitute the base isolation layer of the base‐isolated structure. A refined finite element analysis is carried out to accurately reveal the dynamic characteristics of the base‐isolated structure. It is shown that the base‐isolated structure can be well simulated by the finite element model, which is verified through a comparative study with the results of shaking‐table tests. The base‐isolated structure gains a sound seismic mitigation compared to the base‐fixed structure in a probabilistic sense, and can ensure the safety of the structure by deploying a moderately designed base isolation system.

Evaluation of the seismic site response in randomized velocity profiles using a statistical model with Monte Carlo simulations

Nonlinear stochastic seismic dynamic response analysis of submerged floating tunnel subjected to non-stationary ground motion

Probability density evolution method for seismic displacement-based assessment of earth retaining structures

An extended two-scale random field model for stochastic response analysis and its application to RC Short-leg shear wall structure

NaTech events define the interaction between natural hazards and industrial accidents leading to major fires, explosions or toxic releases where hazardous substances are involved. Among NaTech events, earthquake is one of the most important, because it affects the entire plant and can cause simultaneous damage to different equipment. This study evaluates the local seismic response at the Bussi MHIP Chemical Company, located in Abruzzo region (Central Italy). It is a representative Major-Hazard Industrial Plant (MHIP) subject to Italian standard Decree (D.Lgs. 105/2015; Directive 2012/18/EC - Seveso III), which requires a multidisciplinary approach, given the high complexity of the problem and the numerous types of equipment. We focus on the EURECO plant, located inside the Bussi MHIP, which is featured by high seismicity with potential seismic amplification phenomena due to its complex geo-lithological setting. In the present framework, the influence of soil profile properties under the EURECO plant is investigated through a stochastic site response analysis. We aim to conduct a sensitivity analysis to assess the amplification factor (AF) variability with the randomness of soil properties and geological setting. The implementation of a geological reference model was supported by building a geo-database based on 125 collected boreholes stratigraphies, which identify the main lithological units and their spatial relationships. The limited availability of deep boreholes led to the adoption of other 110 virtual boreholes, where lithological column was extrapolated from stratigraphic sections, geological maps and information from the literature. This approach allowed us to integrate and combine actual with virtual subsurface data, through expert interpretation. Soil properties were collected from a review of relevant literature and previous Seismic Microzonation studies of geologically compatible areas. From this collection, we derived physical (e.g., g, Vs) and dynamic (e.g., shear modulus G/G0, damping D) properties for each soil type. In this study, the shear modulus and damping curves proposed by Darandeli (2001) and modified by Gaudiosi (2023) were applied to each soil type. The geotechnical properties and the variability associated with their distributions have a high impact on the seismic response of a site.  From the geological model, we defined the range of variability of the parameters associated with each seismic unit (e.g. Vs shear waves, bedrock depth, geomechanical properties). We assumed two scenarios in the ultimate conditions of the plant, the Safe Life State (SLV) and the Collapse Limit State (SLC) according to the National Building Code (NTC2018). The seismic inputs were selected using the Probabilistic Seismic Hazard Analysis (PSHA) approach. We performed numerical simulations of 1D – 1D stochastic -2D site response to take into account the influence of the variability of soil parameters and selected seismic input on amplification factor (AF). The development of a seismic response analysis for each simulation allowed us to calculate the AF for the EURECO plant within the estimated fundamental period of vibration of a specific H202 storage tank located inside it. The simulated seismic scenarios could involve the overturning of the storage tank, leading to fires or the release of toxic substances. 

Comparison of PDEM and MCS: Accuracy and efficiency

Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures

Seismic assessment is essential to maintain pile-supported wharves (PSW) functionality and limit earthquake losses. However, existing studies have not fully addressed the stochastic dynamic response of PSW systems and the evolution of their time-dependent seismic reliability. To address this, a Probability Density Evolution Method (PDEM)-based framework is developed to quantify reliability degradation under stochastic excitation. The results indicate pronounced non-Gaussian evolution of the response probability density over time, including broadening, peak reduction, skewness, and multimodality. Consistently, the system reliability exhibits a non-smooth, stepwise decline, implying that the underlying response process deviates from conventional Poisson or Markovian assumptions.

Predicting ground motion over large areas seems to be the frontier in the enhancement of seismic hazard maps at the national level. Simplified approaches based on 1D numerical simulations have gained traction due to the limited availability of detailed geological and geotechnical data. The Italian Project PRIN-SERENA (mapping Seismic site Effects at REgional and National scAle) aims to improve ground motion amplification maps by deriving site-specific amplification functions (FAs) through numerical modelling and validating these models against experimental data to ensure reliability and practical applicability.In the framework of WP6-SERENA project, we evaluated the suitability of these simplified methods in 21 seismic stations located an intermontane basin in the southern Apennines (Val d’Agri, southern Italy), characterised by high seismicicity and complex geology. The amplification functions (FA) were experimentally estimated by applying a nonparametric single-step generalized inversion (Klin et al., 2018) to a database of about 2000 waveforms from local and regional events (up to 400 km), providing robust azimuthal coverage for each station.For the 21 sites experimental FAs were estimated for three period bands of engineering interest: 0.1 – 0.5 s, 0.4-0.8 s and 0.7-1.1 s. These estimates were compared with those derived from the 1D stochastic modelling results by using the NC92soil software (Acunzo et al., 2024) and the subsurface seismo-stratigraphic and mechanical models as input data. The modelled FAs exhibited a narrow amplification range (0.7-1.5), while experimental FAs showed a broader range (1-5). Sites that have the same stratigraphic and mechanical subsurface model have identical modelled FAs, while experimental FAs have very different values reflecting the actual geological complexity of the sites. This discrepancy is most evident at sites with thick surface soils (>100 m), where the 1D models were unable to fully account for subsurface conditions.For the sites with pronounced discrepancies, deterministic modelling incorporating detailed subsurface characteristics and a higher shear velocity contrast between sediments and bedrock was performed. For sites with low sediment cover ( 200 m), deterministic modelling could not reproduce the experimental findings.These results highlight the limitations of simplified 1D modelling, which primarily accounts for stratigraphic amplification but fails to capture the full complexity of the local seismic response, influenced by path and source effects, as the experimental approach does. Simplified numerical estimates of FAs risk underestimating or misrepresenting seismic site responses, particularly in geologically complex settings such as valleys and intermontane basins. This study underscores the importance of integrating experimental data into seismic hazard assessments to account for the complexity of local seismic responses. Acunzo, G. et al. 2024. NC92Soil: A computer code for deterministic and stochastic 1D equivalent linear seismic site response analyses. Computers and Geotechnics, 165, p.105857.Klin, P. et al. 2018. GITANES: A MATLAB package for estimation of site spectral amplification with the generalized inversion technique, Seismol. Res. Lett. 89, no. 1, 182–190.

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

Running trains on a long-span bridge under earthquake is an inevitable problem for the high-speed railway in seismic regions. This paper aims to investigate the seismically random dynamic behavior of train–cable stayed bridges interaction system using the probability density evolution method (PDEM). The system simultaneously involves the multi-effect of random multi-point earthquake excitations with traveling waves and random track irregularity. The multi-point earthquake excitations, composed of lateral and vertical positions, are generated by stochastic harmful functions (SHFs) and uniformly modulated into nonstationary random processes. The motion equation of such a system is established by coupling the 38-degree vehicles and bridges through the refined random wheel/rail contact relationship and accounting for the phase-lags of seismic travelling waves between pier excitations. PDEM is proven to be applicable to such time-dependent systems and is then used to transform the random excitations into a series of deterministic representative excitations with initial probability. By solving for the corresponding deterministic probability responses, various nonstationary random responses, including the time-dependent probability density evolution functions of random responses, mean values curves and standard deviations, can be obtained easily. A case study based on the cable-stayed bridge is then presented, and the results show the effectiveness and accuracy of the proposed method by comparison with Monte Carlo Simulation. Additionally, the influence of train speed and seismic intensity on the safety and reliability of trains is discussed.

Probability density evolution method based stochastic simulation of near-fault pulse-like ground motions