Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions

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

Ground seismic response analysis based on the probability density evolution method

Comparison of PDEM and MCS: Accuracy and efficiency

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

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

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.

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

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

A family of simulation methods for nonstationary earthquake ground motions is proposed. It employs a univariate model of phase spectrum built up on a time argument associated with the concept of starting-time of phase evolution of frequency components. This phase model allows a feasible phase spectrum just using few variables of the starting-time in numerical implementation. In order to reduce the computational effort of the starting-time, a wave-group propagation formulation is also introduced. Two observed ground motions at the type-II site, i.e. Northridge and Chi-Chi waves, are investigated for illustrative purposes. Inspired from the proposed method, a numerical technique for the spatial variation of ground motions is also developed, and the investigation of coherency function between the example ground motions observed at different stations is carried out. Numerical results prove the validity and applicability of the simulation scheme. This methodology provides a new perspective towards the representation of nonstationary stochastic ground motions.

Point group identification algorithm in dynamic response analysis of nonlinear stochastic systems

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

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.

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.