Narrow Results

The intrinsic relationship between deterministic system and stochastic system is profoundly revealed by the probability density evolution method (PDEM) with introduction of physical law into the stochastic system. On this basis, stochastic dynamic stability analysis of single-layer dome structures under stochastic seismic excitation is firstly studied via incorporating an energetic physical criterion for identification of dynamic instability of dome structures into PDEM, which yields to sample stability (stable reliability). However, dynamic instability is not identical to structural failure definitely, where strength failure can be experienced not only in the stable structure but also when the structure is out of dynamic stability. It is practically feasible to decouple the stochastic dynamic response of dome structures to be a stable one and an unstable one according to the generalized density evolution equation (GDEE). Consequently, the global failure probability can be investigated separately based on the corresponding independent stochastic response. For unstable failure probability assessment, the failure probability is the unstable probability if the dome's failure is attributed to instability, whereas inverse absorbing is firstly implemented to get rid of the stochastic response before instability and a complementary process is filled in the safe domain immediately to finally assess the probability of strength failure after dynamic instability.

The probability density evolution method (PDEM) provides a feasible approach for the dynamic response analysis of nonlinear stochastic structures. The key step in this regard is to solve a generalized density evolution equation (GDEE) in order to establish the probability density function (PDF). Previously, a finite difference method (FDM) has often been resorted to solve the GDEE. However, one may encounter the problem of mesh sensitivity in the application of FDM to the PDEM. To this end, a novel difference-wavelet method that can improve the finite difference result by means of a nonlinear wavelet density estimation method is proposed in the present paper. By exploiting the multi-resolution property of wavelet functions and by choosing the optimal scale at each instant, it is expected that the bothering mesh sensitivity issue in finite difference method can be overcome to some extent and a better probability density result can be obtained. In order to verify the proposed method, a single-degree-of-freedom (SDOF) oscillator and an 8-story frame structure are investigated in detail. The results show the notable superiority of the proposed method to finite difference method.

Stochastic dynamic analysis of structures with random parameters continues to be an open question in the field of civil engineering. As a newly developed method, the probability density evolution method (PDEM) can provide the probability density function (PDF) of the dynamic responses of highly nonlinear structures. In this paper, a new method based on PDEM and the kriging surrogate model, named the K-PDEM, is proposed to study the stochastic response of a structure. Being an exact interpolation method, the Gaussian process regression or the so-called kriging method is capable of producing highly accurate results. Unlike the traditional PDEM numerical method whose numerical precision is strongly influenced by the number of representative points, the K-PDEM employs the kriging method at each instant to generate additional time histories. Then, the PDEM, which is capable of capturing the instantaneous PDF of a dynamic response and its evolution, is employed in nonlinear stochastic dynamic systems. Because of the decoupling properties of the K-PDEM, the numerical precision of the result is improved by the enrichment of the generalized density evolution equations without increasing the computation time. The result shows that the new method is capable of calculating the stochastic response of structures with efficiency and accuracy.

The systematic running safety assessment of railway bridges presents lots of challenges, one of which is estimating the uncertainty bounds of the structural responses of bridges under vehicle loads with multisource randomness. In this study, a probability safety assessment method is proposed for evaluating the uncertainty bounds of random time-history responses for the stochastic train-bridge coupled system. First, a refined probabilistic model for the train-bridge coupled system (TBS) in heavy haul railway is established with the multi-excitations of random track irregularities, random vehicle loads and stochastic structural parameters. The probability density evolution method (PDEM) is employed to obtain the solution of the time-varying probability transferred between the stochastic excitations and the output of the dynamic responses. Then, to establish a rapid and straightforward approach for the systematic running safety assessment of the TBS, the quantiles of the probability distribution are used to estimate the time-history uncertainty bounds of random responses of interest distributed in real probability functions. Case studies by the field test and numerical simulation are presented to verify and investigate the accuracy and reliability of the proposed method. The results show that the quantiles of the probability distribution proposed are suitable for the systematic running safety assessment of the TBS.

The high-speed maglev vehicle-guideway coupled system (MVGCS) is a complex system, whose random vibration characteristics have not been well studied due to a limited number of examples. To address this issue, a new efficient approach is proposed for the random vibration analysis of the MVGCS, which combines the probability density evolution method and multi-time step method with multiple random loads considered. The random model established for 10-degree-of-freedom maglev vehicles and guideway is time-dependent, considering two different supporting conditions. The Monte Carlo method is used to assess the accuracy and efficiency of the proposed approximate approach, and the random model is verified through comparison with available results. The stochastic dynamic responses of the vehicles, guideway, and electromagnetic levitation forces, including the mean values and standard deviations, are determined in a case study. The results show that the proposed method is feasible for the dynamic analysis of maglev systems with a reasonably good efficiency in computation. Furthermore, critical parametric analyses involving vehicle speed, irregularity, and cut-off wavelength are performed with the results discussed.

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.

The dynamic performance of high-speed maglev trains, a next-generation rapid transit system, has received continuous attention in recent years. In this study, a dynamic reliability analysis method for a high-speed maglev train–guideway coupled system was proposed. First, a refined model of the maglev vehicle–bridge interaction system was established, where the vehicle subsystem was simulated as a rigid body-spring-damper model with 101 degrees of freedom. The guideway subsystem was simulated as a finite element model, and these two subsystems were coupled as an entire system through a magnet–rail interaction model with a proportional-derivative (PD) controller. Second, a dimension-reduction method for the simulation of representative samples of track irregularities was developed, and thus the number of random variables in the system was reduced to four. Finally, an efficient method for the calculation of the dynamic reliability of a maglev train–guideway coupled system was proposed using the probability density evolution method-based equivalent extreme value principle. With numerical examples, the accuracy of the maglev train–guideway interaction model was verified by comparing it with field measurement data from the Shanghai high-speed maglev line. The accuracy of the proposed dynamic reliability analysis method was confirmed by comparing three types of results, that is, the mean value time-history curve, the probability density function, and the cumulative distribution function of extreme values, obtained by the Monte Carlo method. Finally, the dynamic reliability of the running safety and stability of the maglev vehicle at a speed of 430 km/h and the variation laws of the dynamic reliabilities with train speed were examined in detail.

The method to protect the whole station structure by installing flexible devices at the column end has been proved to be effective in typical station structures. However, the damping effects and application of this method to multi-story subway station structures still need further study. In this paper, a stochastic analysis method, probability density evolution method (PDEM), is adopted to estimate the damping effect of a four-story subway station structure with lead rubber bearings (LRBs) installed in the central column. Stochastic analysis results show that the internal forces and plastic energy dissipation of columns installed LRBs are greatly reduced and the reliability is improved. Internal forces and plastic energy dissipation of columns without the installation of LRBs increase and the reliability decrease. There is little difference in structural deformation and plastic energy dissipation of structural components when LRB is installed in the middle of the column or at the column ends. When LRBs are installed only in some of the stories, the overall reliability of the structure is lower than that of the structure without LRB. When LRBs are installed in the columns of all stories, the structural reliability is higher than that of the structure without LRB. Based on the above results, to obtain better seismic control effect by installing LRBs, the multi-story subway station structure should be installed with LRBs in the columns of all stories, and LRBs can be installed at the upper, middle and lower end of the columns.

A stochastic function model of seismic ground motions is presented in this paper. It is derived from the consideration of physical mechanisms of seismic ground motions. The model includes the randomness inherent in the seismic source, propagation path and local site. For logical selection of the seismic acceleration records, a cluster analysis method is employed. Statistical distributions of the random parameters associated with the proposed model are identified using the selected data. Superposition method of narrow-band wave groups is then adopted to simulate non-stationary seismic ground motions. In order to verify the feasibility of the proposed model, comparative studies of time histories and response spectra of the simulated seismic accelerations against those of the recorded seismic accelerations are carried out. Their probability density functions, moreover, are readily investigated by virtue of the probability density evolution method.