Probabilistic Engineering Mechanics最新文献

{"title":"A general and efficient time-domain random fatigue analysis method under non-stationary non-Gaussian random excitations","authors":"Sisi Peng,&nbsp;Wenliang Fan,&nbsp;Zhisong Wang,&nbsp;Zhengliang Li,&nbsp;Bo Yang","doi":"10.1016/j.probengmech.2026.104008","DOIUrl":"10.1016/j.probengmech.2026.104008","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 104008"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148737195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Seismic failure probability assessment method for concrete gravity dams based on near-fault ground motion parameter inversion","authors":"Tuodi Liu,&nbsp;Zhiqiang Song,&nbsp;Chuang Li,&nbsp;Xiangjiang Li,&nbsp;Zongkai Wang,&nbsp;Yunhe Liu,&nbsp;Zhilong Li","doi":"10.1016/j.probengmech.2026.104012","DOIUrl":"10.1016/j.probengmech.2026.104012","url":null,"abstract":"<div><div>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 (<em>T</em>_p), peak ground velocity (PGV), moment magnitude (<em>M</em>_<em>w</em>), and epicentral distance (<em>R</em>) were fitted in this study. Considering the randomness of ground motion parameters including <em>M</em>_<em>w</em>, <em>R</em>, 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 <em>M</em>_<em>w</em>, <em>R</em>, 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.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 104012"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148776945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reliability of multi-state weighted k-out-of-n systems with shared frailty and stochastic thresholds","authors":"Murat Ozkut,&nbsp;Hao Lyu","doi":"10.1016/j.probengmech.2026.104016","DOIUrl":"10.1016/j.probengmech.2026.104016","url":null,"abstract":"<div><div>We study the reliability of heterogeneous multi-state weighted <span><math><mi>k</mi></math></span>-out-of-<span><math><mi>n</mi></math></span> systems whose components operate in a shared random environment. Existing models typically assume a fixed system threshold or treat a random threshold as independent of component degradation. However, an environmental regime that accelerates degradation may also be associated with a higher scheduled delivery requirement. We capture this feature through a shared frailty variable that simultaneously modulates the component degradation rates and the system threshold. Conditional on the frailty, components are independent, while successive sojourn times within each component are dependent through FGM copulas. Closed-form component state probabilities are derived for three-state and four-state components, and the system reliability is computed by a finite multinomial summation over admissible configurations. The marginal reliability follows by averaging over the frailty distribution. We also establish monotonicity properties, define a rate-based reliability importance measure, and introduce a decoupled benchmark that quantifies the coupling penalty caused by treating component degradation and the system requirement as independent. A numerical validation example and a wind farm case study show that ignoring the shared degradation-demand mechanism can overestimate reliability. The frailty distribution and the threshold schedule are the main drivers of reliability.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 104016"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148951227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A functional integral formulation for stochastic engineering mechanics","authors":"Ioannis A. Kougioumtzoglou","doi":"10.1016/j.probengmech.2026.103987","DOIUrl":"10.1016/j.probengmech.2026.103987","url":null,"abstract":"<div><div>A versatile and computationally efficient functional-integral-based methodology is developed for the stochastic response determination of nonlinear engineering mechanics systems, in which uncertainty may arise either as additive excitation or through stochastic operators embedded within the governing equations. The formulation relies on the concept of the probability density functional, representing the infinite-dimensional analogue of the conventional probability density function (PDF), thereby enabling a continuum-level probabilistic description of stochastic fields without <em>a priori</em> discretization. Further, relying on this representation of stochastic fields, a general probabilistic–variational framework is developed, wherein the governing equations are enforced as functional constraints within a functional integral representation of the response PDF. Furthermore, the functional integral is approximated via a variational principle, based on the extremization of an appropriately defined stochastic action functional, yielding the most probable admissible field configuration. The methodology exhibits several significant advantages: a) the complexity of the original stochastic problem is substantially reduced, as it is recast as a standard deterministic boundary value problem for the most probable field configuration; b) the formulation enables the direct determination of the full response PDF, rather than being restricted to low-order statistical moments; c) both additive stochastic excitation and stochastic operators of general form (including nonlinear and nonlocal) can be treated in a unified and direct manner, while the classical Wiener path integral formulation is recovered as a special case when an explicit input–response transformation is available; and d) the framework provides natural localization capabilities, enabling the direct evaluation of marginal PDFs for response quantities of interest at prescribed spatial locations. The accuracy of the methodology is assessed through both a linear benchmark problem, for which exact solutions are recovered, and a geometrically nonlinear, statically indeterminate stochastic beam problem, where comparisons with Monte Carlo simulations demonstrate satisfactory agreement. Overall, the proposed functional integral formulation provides a fundamentally new pathway for addressing stochastic engineering mechanics problems and a foundation for future developments in stochastic finite element approaches.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103987"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Code-anchored inverse calibration of probabilistic resistance models for structural reliability assessment under sparse data","authors":"Wenfu He,&nbsp;Yuxiang Zhou,&nbsp;Hao Xu,&nbsp;Jie Yang,&nbsp;Kai Wang,&nbsp;Zhexian Chen","doi":"10.1016/j.probengmech.2026.104010","DOIUrl":"10.1016/j.probengmech.2026.104010","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 104010"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148737198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Active learning with heteroscedastic Gaussian Process Regression model","authors":"Ozge Ozbayram,&nbsp;Audrey Olivier,&nbsp;Lori Graham-Brady","doi":"10.1016/j.probengmech.2026.103975","DOIUrl":"10.1016/j.probengmech.2026.103975","url":null,"abstract":"<div><div>Active learning aims to minimize data requirements for accurate surrogate modeling; yet when noise varies across the input space, both the model and sampling strategy must account for heteroscedasticity, which existing approaches often neglect. We present an enhanced Heteroscedastic Gaussian Process Regression (HGPR) framework with active learning for input-dependent noise. First, we introduce an enhancement of an existing HGPR model, leveraging L1 regularization to learn a sparse model for the aleatoric noise. This further enables automatic selection of relevant input features and polynomial degrees within a single optimization run, eliminating computationally expensive cross-validation. Second, we study active learning in this heteroscedastic setting. We adapt and evaluate four acquisition functions (Expected Improvement for Global Fit (EIGF), Active Learning Cohn (ALC), Mutual Information (MI), and Fisher Information (FI)), revealing distinct sampling behaviors. EIGF oversamples high-noise regions, degrading uncertainty calibration; FI clusters samples to refine noise parameters at the cost of mean prediction accuracy; while MI and ALC naturally balance exploration with noise-aware exploitation. Leveraging these complementary behaviors, we propose a hybrid strategy that transitions from EIGF to FI, achieving calibrated uncertainty with interpretable two-phase sampling. Experiments on synthetic functions and a microstructure–property prediction problem show that MI achieves the best overall performance, while the hybrid approach offers comparable accuracy.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103975"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"DR-PDEE-based stochastic dynamical response analysis for high-dimensional nonlinear systems under multiplicative non-white noise excitation","authors":"Ting-Ting Sun,&nbsp;Jian-Bing Chen,&nbsp;Pol D. Spanos,&nbsp;Jie Li","doi":"10.1016/j.probengmech.2026.103968","DOIUrl":"10.1016/j.probengmech.2026.103968","url":null,"abstract":"<div><div>Multiplicative stochastic excitation is important in modeling complex nonlinear dynamics across multiple fields. Further, random noise in physical systems is commonly not ideal white noise. It is noted that stochastic analysis of high-dimensional nonlinear systems, even under additive white noise, has been quite challenging for several decades. These issues are exacerbated in the case of multiplicative excitation and non-white excitation. To address these challenges, the dimension-reduced probability density evolution equation (DR-PDEE) is extended herein to determine the response probability density of such systems. The basic idea is to first augment the original system to a white noise excited higher-dimensional system, and then to reduce the system by the DR-PDEE. In this approach, the multiplicative non-white noise excitation is modelled by an analog filter with white noise input. The original dynamical system combined with the filter then forms an augmented higher-dimensional system driven by additive white noise. Next, the two-dimensional DR-PDEE is constructed for the joint process that combines the quantity of interest in the original system with a filter output, possessing a nonzero intrinsic diffusion function. Solving the DR-PDEE directly yields the probability density function (PDF) of the quantity of interest. Notably, it is found that, owing to the introduction of the analog filter, the original system driven by multiplicative non-white noise is converted into a problem involving additive white noise excitation. Thus, the intrinsic diffusion function in the corresponding DR-PDEE for the joint process becomes a known constant. This feature distinguishes the DR-PDEE for systems driven by multiplicative white noise, where the intrinsic diffusion functions are state-dependent. Thus, only the intrinsic drift functions in the DR-PDEE must be identified. These terms can be reliably estimated through a small number of representative deterministic analyses. An example is presented to validate the proposed method. The numerical example demonstrates that the DR-PDEE yields PDF solutions with high tail accuracy using only a limited number of deterministic analyses.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103968"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uncertainty propagation analysis for non-parameterized probability box based on distance-aware deep learning model","authors":"Jing Ma,&nbsp;Liudong Hou,&nbsp;Zongtai Wang,&nbsp;Dandan Yang,&nbsp;Shenkun Zhao,&nbsp;Bailiang Li,&nbsp;Zhe Zhang,&nbsp;Jinwu Li,&nbsp;Chao Jiang","doi":"10.1016/j.probengmech.2026.103966","DOIUrl":"10.1016/j.probengmech.2026.103966","url":null,"abstract":"<div><div>The uncertainty propagation analysis task for non-parameterized probability boxes (P-boxes) involves a dual-layer nested problem combining probability propagation analysis and interval analysis. Constructing two stage surrogate models serves as the primary framework to address the non-parameterized P-box uncertainty propagation analysis. The inner layer builds an approximate model of the response function concerning input parameters, based on which the outer layer establishes a surrogate model for auxiliary responses corresponding to the bounds of the cumulative distribution function (CDF) for response function with respect to a series of standard uniform distribution variables. This paper proposes a distance-aware deep learning surrogate model-based method for non-parameterized P-box uncertainty analysis to efficiently obtain the upper and lower bounds of the response function's CDF. First, during surrogate model construction, we employ the spectral-normalized neural Gaussian process (SNGP) to characterize the predictive uncertainty of deep learning, which possesses distance awareness, where samples farther from training dataset exhibit greater predictive uncertainty. Second, the deep learning model's predictive uncertainty serves as an active learning function to adaptively identify optimal sample for augmenting the training dataset until stopping criteria are met. Third, the upper and lower bounds of CDF for the response function is computed using Monte Carlo simulation with the outer-layer deep learning model. Finally, two numerical examples validate the effectiveness of the proposed method.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103966"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bayesian updating for Copula-based joint probability modeling of wind speed, wind direction, and air temperature under parameter uncertainty","authors":"Jinxiang Zhang,&nbsp;Mingjin Zhang,&nbsp;Hongbing Liu,&nbsp;Haochao Wang","doi":"10.1016/j.probengmech.2026.103985","DOIUrl":"10.1016/j.probengmech.2026.103985","url":null,"abstract":"<div><div>Accurate estimation of environmental extremes is challenged by limited data availability and complex dependence among environmental variables. To address this issue, a Bayesian updating framework integrating optimal parametric marginal distributions with nonparametric Bernstein Copula-based joint modeling is proposed and validated through a case study. First, the marginal distributions of wind speed, wind direction, and air temperature are identified using a tail-feature-based selection method, and three Bernstein Copula-based joint modeling strategies are developed to characterize nonlinear dependence structures. The model parameters are then dynamically updated through Bayesian inference, and directional environmental contours are constructed using the IFORM approach. The results show that the strategy directly constructing the trivariate joint distribution achieves the best performance. The posterior distributions progressively converge toward the full-data results, accompanied by a reduction in parameter uncertainty. Comparisons with MCMC-based posterior inference further confirm the accuracy of the proposed MAP–Laplace approach. In addition, environmental contours obtained at successive updating stages gradually approached the complete-data results, demonstrating the capability of the proposed framework to provide reliable environmental contour estimates throughout the updating process. Although the maximum wind speeds obtained from different methods are similar, the corresponding temperatures differ by more than 10°C when dependence is ignored. These findings highlight the importance of accounting for both dependence structures and parameter uncertainty in environmental probabilistic modeling and environmental load assessment.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103985"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractional differential equations under Poissonian white noise input: Transient response and probability density function","authors":"Salvatore Russotto,&nbsp;Mario Di Paola,&nbsp;Antonina Pirrotta","doi":"10.1016/j.probengmech.2026.103990","DOIUrl":"10.1016/j.probengmech.2026.103990","url":null,"abstract":"<div><div>Fractional differential equations driven by stochastic excitation are widely used in several engineering and physical applications, as they provide an effective framework for modelling systems characterized by hereditary behavior, memory effects, and nonlocal interactions under random inputs.</div><div>Although the response probability density function (PDF) of fractional differential equations excited by Gaussian white noise has been extensively investigated, this is not the case for fractional differential equations driven by Poissonian white noise. Indeed, while the analysis of integer-order differential equations subjected to Poissonian white noise has been widely addressed, the few available studies concerning fractional differential equations under Poissonian white noise excitation are mainly restricted to the stationary component of the response PDF.</div><div>To overcome these limitations, this paper proposes a continuous-time method to obtain the PDF of the response of two-term fractional differential equations subjected to Poissonian white noise excitation. In the proposed method, the response process is first represented by means of the method of iterated kernels. This representation is then employed to derive the characteristic function of the response process, from which the PDF is evaluated through inverse Fourier transform. To assess the accuracy of the proposed method, numerical simulations have been performed on two different structural systems. The results show an excellent agreement between the PDF obtained through the proposed method and that evaluated by Monte Carlo simulation.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103990"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}