Temperature-induced strain prediction provides a benchmark for structural condition assessment, but existing methods are still limited in quantifying the uncertainty of bridge performance evolution under nonstationary environments. This study develops a framework for bridge temperature-induced strain prediction and uncertainty quantification based on a stochastic variational heteroscedastic Gaussian process (SVHGP). The heteroscedastic relationship between non-uniform temperature field and bridge strain is analyzed. A two-layer probabilistic framework is established that integrates mean-response learning with noise-variance estimation to explicitly characterize input-dependent noise. Following this, the sparse inducing points are introduced for the latent mean and noise functions, respectively. The factorized evidence lower bound (ELBO) is derived to enable mini-batch stochastic variational inference. The variations in epistemic and aleatoric uncertainty are investigated on long-term and short-term time scales. The proposed method was validated using field monitoring data from an in-service concrete cable-stayed bridge under multiple operating scenarios. The results show that the proposed model provides probabilistic predictions of bridge temperature-induced strain and adaptively characterizes input-dependent uncertainty. It also distinguishes epistemic and aleatoric uncertainty across different time scales, while the high-frequency intraday experiment indicates its computational feasibility for nearly million-scale monitoring samples. The proposed model provides an uncertainty-calibrated probabilistic baseline for temperature-induced strain prediction under normal operating conditions.
In this paper, an efficient semi-analytical methodology for the non-stationary stochastic analysis of nonlinear ship rolling in irregular seas is developed. To account for the underlying maritime complexity, the roll dynamics are described by a physically consistent fractional-order system that incorporates added inertia nonlinearity, cubic-quintic restoring characteristics, and nonlinear quadratic viscous damping. A defining feature of the model is the representation of hereditary hydrodynamic damping via a fractional derivative operator, which naturally captures the memory effects inherent in fluid-hull interactions. The non-stationary sea-wave excitation is modeled through a shaping-filter approach, specifically by means of a calibrated fourth-order filter combined with deterministic amplitude modulation, yielding a compact finite-dimensional Markovian representation. The proposed method leverages a refined blend of statistical linearization (SL), harmonic balancing (HB), and a time-varying covariance Lyapunov equation. Its effectiveness rests on a parsimonious yet physically consistent Markovian augmentation of the ship-roll dynamics. Local-in-time nonlinearities are treated via SL, while the fractional-memory contribution is embedded into an evolving equivalent-linear closure through time-dependent HB components. A key methodological refinement involves updating the representative carrier frequency used in the HB step solely from the statistically linearized backbone to avoid implicit coupling. The framework efficiently delivers estimations of time-dependent response variances, evolutionary power spectra, and roll-amplitude probability-density function surfaces. Validation against benchmark Monte Carlo simulations demonstrates a high level of agreement across various sea states and fractional orders. Ultimately, the proposed methodology offers a reliable, physically interpretable, and computationally efficient tool for stochastic ship-roll response and stability assessment in uncertain and dynamically evolving open sea environments.
Classical tuned mass damper (TMD) design assumes precisely known structural properties, but no systematic framework exists when the primary structure is uncertain due to material variability, construction tolerances, or environmental factors. This paper introduces a conceptual dichotomy in robust TMD design: in-sample optimisation, which determines parameters conditionally for individual structural realisations, versus ensemble optimisation, which seeks distributional robustness across the entire population of uncertain structures. This represents a fundamental departure from existing approaches, which treat uncertainty as a post-design consideration rather than embedding it within the optimisation formulation itself. A probabilistic framework is developed in which structural natural frequency is modelled as a random variable, and dynamic responses are characterised through closed-form analytical expressions for mean and variance, validated against Monte Carlo simulation. The in-sample approach applies fixed-point theory to each realisation and aggregates the resulting parameter distributions. The ensemble approach enforces equal-mean constraints on fixed-point frequencies across the uncertainty distribution and minimises response variance. Comparative analyses in both frequency and time domains demonstrate that while conventional deterministic TMDs remain sensitive to detuning, both proposed strategies effectively suppress resonance amplification. The ensemble design achieves superior performance, reducing maximum structural displacement by 21.50% relative to conventional TMD, compared to 19.32% for in-sample design. Beyond parameter optimisation, this work establishes in-sample versus ensemble as a transferable design philosophy for vibration control under uncertainty, with direct implications for adaptive versus mass-produced damper systems in engineering practice.
Evolution problems involving uncertain initial conditions naturally lead to stochastic formulations in which the state variable evolves as a random process. Classical Monte Carlo simulation (MCS) is commonly employed to track the time evolution of associated probability distributions. Still its accuracy requires a very large number of deterministic realizations, resulting in prohibitive computational costs for high-dimensional problems. In the present paper, an efficient method is presented to calculate the evolution of the cumulative distribution function (CDF), even in the presence of many random variables. This approach relies on a one-to-one mapping between the integration domain and a subset of the unit cube, enabling the use of good lattice point sets to evaluate the required integrals with substantially fewer deterministic analyses than MCS. The method is particularly suitable for structural reliability applications, where failure probabilities depend on multiple uncertain parameters. These probabilities may be very small and vary over time due to material degradation, evolving environmental conditions, and non-stationary loads that are sometimes modeled as stochastic processes. Its effectiveness is demonstrated in the context of durability assessment of aging reinforced concrete structures, with a specific focus on the evaluation of the time-dependent probability of corrosion damage under changing climate scenarios.
The Xiaqu concrete gravity dam is located only 5 km from the nearest segment of the Jiali fault zone. The assessment of its seismic failure probability under near-fault conditions depends heavily on the accuracy of the input ground motion. On the basis of historical seismic records from the dam area, empirical relationships among the pulse period (Tp), peak ground velocity (PGV), moment magnitude (Mw), and epicentral distance (R) were fitted in this study. Considering the randomness of ground motion parameters including Mw, R, peak ground acceleration (PGA), and PGV, a near-fault ground motion generation method based on high- and low-frequency superposition was established. On the basis of this method and a combined oblique incidence mode of P- and SV-waves, 1000 finite element seismic response analyses were conducted. Furthermore, a nonlinear mapping between ground motion parameters, oblique incidence angles, and dam displacement responses was established using an intelligent algorithm. By using the results of a specific finite element analysis as monitoring data, the Mw, R, PGA, PGV, and combined oblique incidence angles of the earthquake were determined through inversion. Random ground motions were then generated while preserving the inherent stochasticity of the seismic waves. Finally, the failure probability of the dam under this specific earthquake was assessed using a comprehensive damage index. This was achieved through a variable weight efficacy coefficient method that integrates multiple indices, including the overall damage index, relative displacement at the dam crest, and cumulative sliding at the dam foundation and sliding blocks.
Thermal failure is a critical failure mode in rolling bearings, particularly under high-speed or heavy-load conditions, resulting in severe consequences such as lubrication failure or bearing seizure. However, existing optimization approaches that consider thermal characteristics frequently neglect time-dependent uncertainties, including dynamic loads and ambient temperature variations, which influence thermal failure risk. This study proposes a reliability-based design optimization framework to derive structural parameter designs for rolling bearings that satisfy thermal failure probability constraints. First, a thermal network model is established to capture the bearing's dynamic thermal response using a quasi-static mechanical model combined with a thermal resistance approach. Subsequently, a two-stage time-series sampling method is developed for time-dependent reliability analysis, enabling efficient evaluation of thermal failure probability under dynamic operating conditions. To maximize the bearing's dynamic load capacity, an optimization formulation is introduced that decouples the optimization loop from failure probability estimation. Finally, the effectiveness of the proposed method is demonstrated in a case study involving a deep-groove ball bearing for an electric motor.
Probabilistic characterization of nonlinear stochastic systems remains a central challenge in science and engineering, particularly when the response exhibits strong non-Gaussian and multimodal characteristics. While moment-based approaches provide an efficient alternative to solving high-dimensional Fokker–Planck equations, their application to nonlinear systems is hindered by the well-known closure problem, for which existing methods, such as the cumulant-neglect closure method, often fail in the presence of bistability. This study presents a novel nonlinear closure framework, termed the Oscillatory Cumulant Closure (OCC) method, for resolving stochastic systems with bistable potentials. The approach is formulated directly within the framework of cumulant equations, enabling higher-order cumulants to be expressed as structured functions of lower-order ones rather than being neglected. The method is developed by interpreting the characteristic function as the Fourier transform of the response probability density, thereby revealing an inherent oscillatory structure in the generating function of bistable systems. By approximating this structure as a harmonic component modulated by a decaying envelope, the OCC method derives explicit relationships for higher-order cumulants governed by a characteristic parameter related to the dominant response modes. This formulation yields a closed system of cumulant equations coupled with an additional algebraic constraint that is solved simultaneously. The proposed method is validated on both symmetric and asymmetric bistable Duffing oscillators over a broad range of nonlinearities and noise intensities. The results demonstrate that the OCC method accurately predicts stationary response statistics, even at relatively low closure levels and under strong stochastic excitation, substantially outperforming the classical CNC approach.
The dependability of downhole tool systems is paramount for the safe and effective execution of offshore oil and gas operations; however, the offshore testing phases for these tools are characterized by intricate hazards, where inadequate management can precipitate catastrophic incidents. This study proposes an integrated risk assessment framework that couples Failure Mode and Effects Analysis (FMEA) with Multi-State Bayesian Networks (MSBN), enabling the systematic identification, quantitative evaluation, and prioritization of potential failure modes during the sea trial phase of ultra-high-temperature and high-pressure downhole tools. A multi-source data-driven analytical system was developed, synthesizing historical failure records, structured expert knowledge, and quantitative risk models. The results precisely delineate high-risk failure modes and their potential consequences, while also providing targeted risk mitigation measures encompassing both technical enhancements and management optimizations. The methodologies and practical strategies presented herein establish a scientific foundation for enhancing the safety and reliability of offshore downhole tool testing. This work holds significant theoretical and practical value for operational risk reduction, the prevention of environmental disasters, and the promotion of sustainable marine oil and gas resource development.
Load-sharing performance is a critical indicator for the transmission system of wind turbine gearboxes, as enhancing it effectively reduces structural vibration and noise, improving operational reliability, and extending service life. However, this performance is susceptible to multi-source uncertainties in practical engineering, such as manufacturing and installation errors, tooth thickness variations, and backlash. Given the limitations of traditional deterministic design optimization in addressing these uncertainties, this paper proposes a reliability-based design optimization (RBDO) framework to improve the load-sharing performance of wind turbine gearboxes. Firstly, a dynamic model for the internal meshing mechanism of a wind turbine gearbox is developed and validated against a load-sharing test. Subsequently, an RBDO model is established considering multi-source uncertainties. To accurately evaluate the reliability of the load-sharing performance of wind turbine gearboxes, an improved step length adjustment (iSLA) method based on the performance measure approach (PMA) is proposed, and its effectiveness is demonstrated through comparative analyses with existing methods. Furthermore, within the proposed RBDO framework, a Kriging surrogate model is adopted to replace the computationally intensive dynamic model of the wind turbine gearbox to enhance computational efficiency, and the Sobol’ method is utilized to identify key uncertain parameters influencing the load-sharing performance. The optimization results indicate that the reliability of the load-sharing performance increases from 0.9061 to 0.9990, and optimized tolerance allocation is achieved. This study provides an effective solution for the reliable design of load-sharing for wind turbine gearboxes.
Probabilistic resistance models for existing structures should reflect both statistical uncertainty and the design-code assumptions under which the structures were originally designed. This paper develops a code-anchored inverse calibration framework for deriving resistance distributions under sparse member-specific data. The proposed method maps code-level quantities, including the design limit, code-implied margin, coefficient of variation and constraint class, into distributional parameters without post hoc sample fitting. Two anchoring modes are distinguished according to the meaning of the code limit. For variables with a physical lower bound, a hard-lower-bound shifted gamma model is calibrated to ensure admissible support. For composite structural resistance variables, the design demand is treated as a lower-tail reference so that the failure-relevant region is preserved. Calibration equations are formulated, and the admissibility, uniqueness and continuous dependence of the hard-bound calibration root are discussed. The framework is verified using reinforced-concrete beam flexural resistance and concrete cover thickness. In the beam example, the zero-fitting anchored model reproduces the Monte Carlo resistance distribution with a Kolmogorov-Smirnov distance of 0.011 and gives a close lower-tail failure probability. In the cover-thickness example, the shifted gamma model respects the code minimum, whereas an unconstrained normal model assigns about 10% probability to inadmissible values. The results show that the proposed framework can provide a transparent initial probabilistic model for sparse-data reliability assessment.
Due to the complex work conditions of rolling bearings, the vibration signals generated exhibit nonlinear characteristics, which means that single-scale feature extraction techniques are hard to extract fault features accurately. Despite its application in mechanical engineering, multiscale fluctuation dispersion entropy (MFDE) is hindered by an inadequate coarse-graining process, which leads to poor stability and large errors. Consequently, a novel multiscale composite optimization fluctuation dispersion entropy (MCOFDE) is proposed for feature extraction. This MCOFDE designs a novel slippage-averaging multiscale approach for signal coarsening, accounting for the connection between data before and after breakpoints, thereby reducing loss of key message in the entropy value. The proposed MCOFDE has been validated through simulation experiments. Comprehensive experiments under varying training ratios, scales, and classifiers demonstrate that MCOFDE outperforms classical methods in diagnostic accuracy and reliability.
We investigate the optimal design of both conventional and non-traditional tuned mass dampers (TMDs) subjected to narrowband random excitation, with the objective of minimising the kinetic energy of the primary structure. A key contribution of this work is the development of a novel analytical formulation for the narrowband excitation case, yielding exact steady-state second-order moment equations for the conventional TMD. Although these expressions can, in principle, be written in closed form, they are prohibitively lengthy and do not lead to practical tuning rules. Instead, the exact moment equations are solved numerically to evaluate the mean-square velocities of the primary and secondary masses, providing a direct and accurate measure of their kinetic energy. The analysis demonstrates that, under narrowband stochastic excitation, the optimal tuning deviates fundamentally from classical deterministic results and does not coincide with existing optimal solutions. To identify optimal parameters, three optimisation strategies, reinforcement learning (RL), particle swarm optimisation (PSO), and surrogate-based Kriging optimisation (SO+KRG), were systematically benchmarked and validated under both deterministic and stochastic conditions. Among these, reinforcement learning consistently delivers superior performance in terms of accuracy and computational efficiency, particularly as the dimensionality of the design space increases. Moving beyond classical configurations, this study presents a comprehensive exploration of non-traditional TMD topologies, which have not previously been examined in a unified framework. An automated incidence-matrix approach was implemented to generate all admissible spring–damper network configurations connecting the primary and secondary masses. From a set of 29 candidate topologies with up to two springs and two dampers, several non-conventional layouts were identified that outperform the classical TMD across both deterministic and stochastic regimes. The optimisation of topologies reveals a common structural characteristic shared by the most effective absorbers, providing a simple yet robust design principle for next-generation TMD systems operating in deterministic and stochastic environments.
In the Karhunen–Loève (KL) series expansion framework for random field discretization, computational errors in common KL methods (e.g., the orthogonal series expansion (OSE) and Jacobi–Legendre–Galerkin (JLG) methods) arise not only from the truncation error due to the number of retained terms (M), but also from eigensolution accuracy related to the number of Gauss integration points (N1D GP) and basis function terms (N1D). The OSE method with a full tensor-product construction and the JLG method with total-order truncation were employed for separable and nonseparable autocorrelation functions, respectively, thereby revealing the decoupled effects of the number of N1D GP and the number of N1D on the eigenvalues and eigenfunctions. The results indicate that the global covariance error may not fully capture local reconstruction errors within the physical domain. For N1D ≥ M + 2, N1D GP primarily influences eigenvalues (λ), while N1D dictates fluctuations in eigenfunctions (f(x)). Accordingly, parameter selection criteria are established for one-dimensional and multi-dimensional random fields, respectively. Engineering case studies indicate that eigensolution accuracy affects the range of variation in load-settlement curves for shallow foundations and that the tilting failure mode of a pile group is particularly sensitive to it. Insufficient accuracy of the eigensolution causes drifts at the most unfavorable failure points and shifts in parameter sensitivity rankings within pile group systems. Furthermore, the convergence pattern of eigensolution errors directly maps onto the convergence behavior of the reliability assessment for pile group systems. This research provides a vital theoretical basis and accuracy parameter selection benchmarks for geotechnical reliability analysis within the KL framework.
Fatigue damage assessment under complex non-stationary non-Gaussian (NS-NG) random loadings remains a challenging problem in structural engineering, as existing time- and frequency-domain fatigue analysis methods suffer from inherent limitations in accuracy and applicability. This study develops a generalized, high-efficiency time-domain random fatigue analysis method for structures under NS-NG excitations. The Johnson transformation model and a sample-interpolation-based technique are introduced to realize accurate and fast generation of NS-NG random excitations. Inspired by the explicit time-domain method (ETDM), an explicit time-domain expression of hotspot stress for linear time-invariant structures is derived, and an efficient equation-solving strategy is proposed to conveniently determine the unknown coefficient matrices. Based on the obtained stress responses, structural fatigue damage is evaluated using the rainflow counting algorithm and cumulative damage criteria, and the statistical characteristics of fatigue damage can be determined by the Monte Carlo simulation. Numerical comparisons with different time- and frequency-domain methods demonstrate that the proposed method provides an accurate, efficient and broadly applicable solution for fatigue analysis under complex random loadings.
