Uncertainty propagation of flutter analysis for long-span bridges using probability density evolution method

An ensemble evolution numerical method for solving generalized density evolution equation

Comparison of PDEM and MCS: Accuracy and efficiency

In civil and offshore engineering, the safety and serviceability of complex structures are of great concern due to the randomness involved in both external excitations and structural properties. Therefore, nonlinear stochastic dynamics of multi-dimensional complex systems plays a central role in the performance assessment of such structures. In the past decade, a new family of probability density evolution method (PDEM) was developed for this purpose. In this method, based on the principle of preservation of probability, a generalized density evolution equation was derived. In contrast to the celebrated FPK and Liouville equations, the dimension of this family of equation depends only on the number of quantities of interest instead of the dimension of the original dimension. In this paper, a newly proposed generalized F-discrepancy (GF-discrepancy) is outlined and adopted as the objective function in point selection of PDEM. By doing so, the efficiency and accuracy of PDEM could be improved and guaranteed for random variables of general distributions. Numerical examples are illustrated. Problems to be further studied are discussed.

Dynamic reliability analysis of Aerial Building Machine under extreme wind loads using improved QBDC-based active learning

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

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.

Limit analysis of stochastic structures in the framework of the Probability Density Evolution Method

A novel method is proposed in this work for the assessment of the remaining fatigue life and fatigue reliability of concrete bridges subjected to random loads. The fatigue reliability of a bridge is a function of the fatigue damage accumulation; a stochastic fatigue damage model (SFDM) with physical mechanism is introduced for deriving the fatigue damage process. In order to implement the probabilistic analysis, based on the probability density evolution method (PDEM), the generalised density evolution equation (GDEE) for the remaining fatigue life is developed. Finally, a prestressed concrete continuous beam bridge located in China is illustrated. The random fatigue load acting on the bridge is modelled as the compound Poisson process, and the simulation of the random load uses the stochastic harmonic function (SHF) method. To simplify the reliability analysis, an equivalent constant-amplitude (ECA) load process is introduced based on energy equivalence. By employing SFDM, the finite element analysis of the bridge under the fatigue loading is performed. Then, the fatigue damage accumulation process of the bridge under the fatigue loading is obtained. Through solving the probability density evolution equation for the remaining fatigue life, the probability density functions (PDFs) of the remaining fatigue life evolving with time is obtained. The fatigue reliability is then calculated by integrating the PDF of the corresponding remaining life.

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

A PDEM based new methodology for stochastic dynamic stability control of nonlinear structures with fractional-type viscoelastic dampers

In the present paper, a framework of dimension-reduction modeling method is developed for a dual stochastic dynamic structural system of spectrum-compatible non-stationary stochastic ground motion processes and stochastic structures. With the aid of the proposed method, the random variables used to describe the stochastic characteristics of the non-stationary ground motion processes and the structural parameters are respectively represented by the one-elementary-random-variable functions, contributing to an entire stochastic dynamic structural system readily described by merely two elementary random variables. Owing to the fact that the number of the elementary random variables needed is extremely small, the set of the representative points associated with the elementary random variables can thus be selected by the widely-used number theoretical method, and then the probability density evolution method can be conveniently combined to conduct the dynamic response analysis and dynamic reliability evaluation of nonlinear stochastic structures. In the numerical examples, the probability density evolution analysis of an eight-storey reinforced concrete frame structure with random parameters subjected to spectrum-compatible non-stationary stochastic ground motion processes is investigated as a case study. Numerical results fully demonstrated the effectiveness and robustness of the proposed method.

The probability density evolution of structural dynamic prediction models under several times random loads are proposed in this paper. Firstly, the probability density of several times stochastic loads can be obtained from the probability density of single time stochastic load. The prediction models of structural reliability from time responses under several times stochastic loads with and without strength degeneration over time are derived using the stress-strength interference theory and probability density evolution method. The differential equation in the models can be solved using the forward finite difference method. The probability density functions of the structural strength redundancy at any time can be obtained. In the end, structural dynamic reliability can be calculated using integral method. The efficiency of the proposed method is demonstrated numerically through a speed reducer. The results have shown that the proposed method is practicable, feasible and gives reasonable accurate prediction.

This article presents an efficient method for analyzing the dynamic reliability of maglev vehicle-bridge coupled systems by combining a theoretical model with an adaptive surrogate model and the probability density evolution method (PDEM). First, a refined theoretical model of a maglev vehicle-bridge coupling system is established. Next, an adaptive surrogate model of the equivalent extreme value of the system dynamic response is established by combining an adaptive sampling method with radial basis functions. Finally, the adaptive surrogate model and PDEM are combined to further improve the efficiency of the dynamic reliability analysis. In the numerical example, the theoretical model of the maglev vehicle-guideway system was first validated by comparing with the measured data from the Shanghai high-speed maglev line. Then, by treating the controller parameters as normally distributed random variables, the accuracy and efficiency of the proposed reliability method were verified through comparison with the Monte Carlo method and the one-stage sampling surrogate model. Additionally, the impact of the randomness of each controller parameter and the coefficient of variation of the controller parameters on the system’s dynamic reliability was discussed.

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.

Ground seismic response analysis based on the probability density evolution method