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In practical engineering, aircraft structures may encounter complex flight environments that can induce intricate stochastic dynamic behaviors, significantly impacting flight safety. However, the analysis of the complex stochastic dynamic behaviors of airfoil structures, especially for fractional viscoelastic systems, is a challenging issue. In this paper, a unified and adaptive approach based on a novel direct probability integral method (DPIM) is proposed to address this challenge. Firstly, the fractional viscoelastic airfoil model under complex flight conditions is established, in which the extreme random flight load is simulated by Lévy white noise. Then, the probability density integral equation (PDIE) of the fractional viscoelastic airfoil system is derived. Furthermore, the important equivalence between PDIE and Einstein-Smoluchowsky differential equation is demonstrated, exhibiting the superiority of PDIE. To solve PDIE, a fully adaptive DPIM without truncating time is presented. The DPIM-based approach is proposed to achieve the global analysis of fractional viscoelastic airfoil systems. The proposed approach provides a new probabilistic perspective for obtaining the generalized stochastic attractor and basin. Moreover, it fulfills stochastic bifurcation and global analyses of the fractional viscoelastic airfoil model under complex flight environments in a unified way. It is worth noting that as the fractional order increases, the stable region becomes more irregular compared to the effects of extreme random flight loads, leading to a rapid decrease in the probability of stability. The generalized stochastic attractor delineates the region of airfoil movement with high flight stability, and the stability probability can be measured by the generalized stochastic basin.

The stationary/nonstationary random vibration responses and dynamic reliability analyses of thin plate structures coupling geometrical nonlinearity and multisource randomness in both structural parameters and external excitations are the important issues to be attacked. In this study, the novel direct probability integral method (DPIM) is proposed to tackle them. Firstly, the differential equations of thin plate vibration with large deflection are revisited based on von Kármán nonlinear theory. Then, the non-intrusive DPIM decouples physical evolution and probability density propagation of structure, which is utilized to calculate nonlinear stochastic dynamic responses and reliability of thin plate in a unified way. Moreover, to determine the resultant large-amplitude nonlinear deformation of thin plate under random concentrated excitation and distributed excitation, an important criterion is devised, which can judge the applicability of geometrically nonlinear theory. Several numerical examples demonstrate the high accuracy and efficiency of proposed approach by comparing the results with those of Monte Carlo simulation. It is indicated that, the changes of structural parameters and random excitation cause the transition of probability density functions of nonlinear deflections from unimodal to bimodal distributions. Stochastic P-bifurcation occurs in nonlinear random vibration analysis of thin plate, which implies that the geometric nonlinearity should be rationally considered in stochastic dynamic analysis of structure. Finally, the remarkable effects of geometric nonlinearity, thickness, random parameter variability and boundary conditions on stochastic responses and reliabilities of thin plates are revealed.

The generalized fractional derivative is a useful tool for describing the mechanical property of viscoelastic dampers in vehicles. However, the complex integral form of the generalized fractional operator and the non-Markov property of fractional system make it challenging to perform stochastic dynamic analysis on multi-degree-of-freedom (MDOF) vehicle systems with generalized fractional damping. Thereby, the main purpose of this paper is to explore the stochastic dynamic characteristics of generalized fractional systems in the framework of the direct probability integral method (DPIM). To this end, a vehicle model with generalized fractional dampers is first constructed to simulate the driving process on uneven pavement, in which the random pavement roughness and discrete impulse excitation are described as continuous Gaussian white noise and discrete Poisson white noise, respectively. Then, the equivalent MDOF stochastic system of the model is derived, and the corresponding probability density integral equation (PDIE) is further established for uncertainty propagation. Benefiting from the historical memory of PDIE, a DPIM-based strategy is proposed to address the random vibration problem of the generalized fractional MDOF system. Finally, the stochastic bifurcation and dynamic reliability analyses of vehicle systems with generalized fractional damping are realized via the proposed strategy. The results indicate that the change of generalized fractional order, as a key design parameter, will cause stochastic bifurcation phenomenon and stochastic chaotic motion, which remarkably affects the vehicle ride comfort and driving safety.

The time-dependent reliability assessment of ballastless tracks in service life is crucial for ensuring the safe and stability of high-speed railways. This work systematically compiles and summarizes the performance functions of ballastless track structures under different failure modes during service period, and then the all modes are regarded as a mixed series-parallel system, a regularization method for performance functions of multiple failure modes is proposed. Subsequently, the direct probability integral method (DPIM) is firstly introduced for evaluating the time-dependent reliability of the CRTS III slab ballastless track system. The accuracy and computational efficiency of the proposed method are verified through comparison with the traditional Monte Carlo method. Furthermore, the failure modes of the system are categorized into safety and applicability. Numerical examples demonstrate the generality and robustness of the developed method in system reliability analysis. The result indicates that environmental factors are the primary cause of diminished system reliability, both in terms of safety and applicability. Extreme temperature gradients are identified as a primary cause of safety performance degradation in track structures, while negative temperature gradients primarily contribute to serviceability deterioration. The mean of train load below 300 kN is essential for maintaining safety performance. The safety and applicability of the track structure after 30 years of service should be a key concern for the railway department. Especially, the reliability calculation method developed in this work for CRTS III systems can be extended to various static and dynamic system state evaluations.

Reliability sensitivity analysis plays an essential role in structural reliability design, especially for reliability-based design optimization (RBDO). However, for some complex and difficult problems, such as the RBDO of time-variant systems with multiple most probable points, there is still a lack of a powerful and versatile method. In this paper, the novel direct probability integral method (DPIM) is proposed for reliability sensitivity analyses to the distribution parameters of random variables and deterministic variables, and the time-invariant and time-variant RBDO problems are resolved in a unified framework. Firstly, in the context of probability density integral, the influences of two types of variables on structural reliability are elaborated. The reliability sensitivities with respect to the distribution parameters of random variables and deterministic variables are derived from probability density integral equation, which significantly improves the computational efficiency and accuracy of reliability sensitivity. Then, the analytical solutions of reliability sensitivity of a typical problem are achieved which are regarded as benchmark solutions. Based on the proposed approach for sensitivity analysis, the time-invariant and time-variant RBDO problems with multiple most probable points can be universally solved. Finally, three examples verify the high accuracy and efficiency of DPIM. It is shown that the proposed approach is capable of addressing RBDO problems with the design variables involving distribution parameters of random variables or/and deterministic variables. Specifically, the numerical results of building structure indicate that the cost of structure increases dramatically with increasing of specified reliability level, and RBDO takes an optimal trade-off between economy and safety in structural design.

Simultaneously determining the random vibration responses and dynamic reliabilities of thin shell coupling geometric nonlinearity and multisource uncertainties is an intractable task. A novel non-intrusive framework based on direct probability integral method (DPIM) is proposed in the present study, which offers an efficient and competitive solution tool to tackle this challenging issue. New framework incorporating DPIM with adaptive schemes can address efficiently stochastic dynamic responses and reliability determination of geometrically nonlinear thin shells in a unified way. Adaptive choosing strategy of the smoothing parameter of Dirac function and the number of representative points is adopted. Importantly, a judgment criterion is established to adaptively perform nonlinear theory and linear theory of large deflection for thin shell, which breaks the limitation of using a single nonlinear or linear theory and results in more accurate responses. Finally, several numerical examples demonstrate that the proposed framework possesses high accuracy and efficiency when compared to Monte Carlo simulation (MCS) and quasi-MCS for computing stochastic deflection and stress responses and dynamic reliabilities. The transform of probability density function of deflection responses from unimodal to bimodal distribution implies that the stochastic P-bifurcation occurs in random vibration of nonlinear thin shell. The remarkable effects of sound pressure level of noise excitation, power spectral density of random excitation, random parameter variability and boundary conditions on uncertainty quantification of thin shells are revealed.

As offshore wind turbines develop into deepwater operations, accurately quantifying the impact of stochastic excitation in complex sea environments on offshore wind turbines and conducting structural fatigue reliability analysis has become challenging. In this paper, based on long-term wind–wave reanalysis data from the South China Sea, a novel direct probability integral method (DPIM) is developed for the stochastic response and fatigue reliability analysis of the key components for the floating offshore wind turbine structures, under combined wind–wave excitation. A 5 MW floating offshore wind turbine is considered as the research object, and a comprehensive analysis of the wind turbine system is performed to assess the short-term fatigue damage at the tower base and blade root. The proposed method’s accuracy and efficiency are validated by comparing the results to those obtained from Monte Carlo simulations (MCS) and a subset simulation (SSM). Additionally, a sensitivity analysis is conducted to evaluate the impact of different environmental parameters on fatigue damage, providing valuable insights for the design and operation of FOWTs in varying sea conditions. Furthermore, the results indicate that the fatigue life of floating offshore wind turbine (FOWT) structures under combined wind–wave excitation meets the design requirements. Notably, the fatigue reliability of the wind turbine under aligned wind–wave conditions is lower compared to misaligned wind–wave conditions.

Based on the Direct Probability Integral Method (DPIM), this study investigates the global seismic reliability of reinforced concrete (RC) frame structures considering the randomness of material parameters and the non-stationarity of ground motions. A doubly non-stationary ground motion model is established using evolutionary power spectrum theory combined with the spectral representation–stochastic function method. A dimensionality reduction technique is adopted to generate ground motion samples compatible with the design response spectrum. A finite element model of the RC frame is developed in Abaqus. Modal analysis and deterministic time history analysis are conducted to obtain the dynamic characteristics and seismic responses of the structure. Based on 600 representative ground motion time histories generated using the maximum frontier (MF) discrepancy sampling method, nonlinear time history analyses are performed. The DPIM is then employed to calculate the statistical characteristics of structural responses and quantify response variability, enabling a rational evaluation of the structural safety margin. Finally, based on the equivalent extreme value event theory and DPIM, the reliability of the structure under a single failure mode and the global reliability under multiple failure modes are computed. The results show that the global reliability of the structure is 82.088%, which is significantly lower than that of any single failure mode. This study provides a quantitative reference for evaluating the global seismic reliability of RC frame structures subjected to nonstationary seismic excitation.

This paper proposes a hybrid algorithm based on the physics‐informed kernel function neural networks (PIKFNNs) and the direct probability integral method (DPIM) for calculating the probability density function of stochastic responses for structures in the deep marine environment. The underwater acoustic information is predicted utilizing the PIKFNNs, which integrate prior physical information. Subsequently, a novel uncertainty quantification analysis method, the DPIM, is introduced to establish a stochastic response analysis model of underwater acoustic propagation. The effects of random load, variable sound speed, fluctuating ocean density, and random material properties of shell on the underwater stochastic sound pressure are numerically analyzed, providing a probabilistic insight for assessing the mechanical behavior of structures in the deep marine environment.

Structural reliability analysis under uncertainty is paid wide attention by engineers and scholars due to reflecting the structural characteristics and the bearing actual situation. The direct integration method, started from the definition of reliability theory, is easy to be understood, but there are still mathematics difficulties in the calculation of multiple integrals. Therefore, a dual neural network method is proposed for calculating multiple integrals in this paper. Dual neural network consists of two neural networks. The neural network A is used to learn the integrand function, and the neural network B is used to simulate the original function. According to the derivative relationships between the network output and the network input, the neural network B is derived from the neural network A. On this basis, the performance function of normalization is employed in the proposed method to overcome the difficulty of multiple integrations and to improve the accuracy for reliability calculations. The comparisons between the proposed method and Monte Carlo simulation method, Hasofer–Lind method, the mean value first-order second moment method have demonstrated that the proposed method is an efficient and accurate reliability method for structural reliability problems.